Intermediate Algebra - 2e, 2017a

Intermediate Algebra 2e
SENIOR CONTRIBUTING AUTHORS
LYNN MARECEK, SANTA ANA COLLEGE
ANDREA HONEYCUTT MATHIS, NORTHEAST MISSISSIPPI COMMUNITY COLLEGE

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HARDCOVER BOOK ISBN-13
B&W PAPERBACK BOOK ISBN-13
DIGITAL VERSION ISBN-13
ORIGINAL PUBLICATION YEAR
10 9 8 7 6 5 4 3 2 1
978-1-975076-49-8
978-1-975076-48-1
978-1-951693-24-4
2020

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openstax.org

n n.
15 ÷ 3 5·3.
m
3.
0.
3.
0.

a b c d
a
b
5+6+2+5=18
c
d
a b c d e
a b c d e
8·9=72, 72 = 8 · 9,
1 · 72, 2 · 36, 3 · 24, 4 · 18, 6 · 12.
1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.
a · b = m,

2·2·2·2·3
6·8,
80.
60.

(2·2·3) (2·3·3) (2·2·3·3).
x, y, a, b, c.

a + b a b a b
a − b a b a b
a · b, ab, (a)(b),
(a)b, a(b)
a b
a b
a ÷ b, a/b, a b , ba a
b
a b;
a b
a = b
a < b a > b
a < b is equivalent to b > a. For example, 7 < 11 is equivalent to 11 > 7.
a > b is equivalent to b < a. For example, 17 > 4 is equivalent to 4 < 17.
a ≠ b
a < b
a ≤ b
a > b
a ≥ b

Parentheses ()
Brackets []
Braces {}
8(14 − 8) 21 − 3[2 + 4(9 − 8)] 24 ÷ ⎧ ⎩13−2[1(6−5)+4] ⎨ ⎫ ⎭ ⎬
Expression Words English Phrase
3 + 5 3 plus 5 the sum of three and five
n −1 n minus one the difference of n and one
6 · 7 6 times 7 the product of six and seven
x
y x divided by y the quotient of x and y
Equation English Sentence
3 + 5 = 8 The sum of three and five is equal to eight.
n −1=14 n minus one equals fourteen.
6 · 7 = 42 The product of six and seven is equal to forty-two.
x =53
x is equal to fifty-three.
y +9=2y −3 y plus nine is equal to two y minus three.
2·2·2·2·2·2·2·2·2.
2·2·2 2 3 2·2·2·2·2·2·2·2·2 2 9 .
2 3 ,
2 3 2·2·2
a n

a n n th
a n “a n th
a 2
a 3
“a squared”
“a cubed”
a 2 a 3
4·2+1
4·2
4·2+1
8+1
9
4+3·7. 4+3
3·7

Parentheses
Exponents
Multiplication Division
Addition Subtraction
Please
Excuse
My Dear
Aunt Sally
18 ÷ 6 + 4(5−2).
30 ÷ 5 + 10(3 − 2).
70 ÷ 10 + 4(6 − 2).

5+2 3 +3 ⎡ ⎣6−3(4−2) ⎤ ⎦.
9 +5 3 − ⎡ ⎣ ⎡ 4(9 + 3) ⎤ ⎦ ⎤ .
7 2 − 2 ⎡ ⎣ ⎡ 4(5 + 1) ⎤ ⎦ ⎤ .
x =4: a x 2 b 3 x c 2x 2 +3x +8.
a

c
x = 3, a x 2 b 4 x c 3x 2 + 4x + 1.
x = 6, a x 3 b 2 x c 6x 2 − 4x − 7.
7, y, 5x 2 ,9a, b 5 .
3x x,
x =1·x.
5x 7 n 2 4 3x 9n 2

7 4
5x 3x
n 2 9n 2
Simplify.
4x +7x + x
Add the coefficients.
12x
2x 2 +3x +7+x 2 +4x +5.
3x 2 + 7x + 9 + 7x 2 + 9x + 8.
4y 2 + 5y + 2 + 8y 2 + 4y + 5.

a
b
a b
a + b
a − b
a · b, ab, a(b), (a)(b)
2a
a ÷ b, a/b, a b , ba
a 14x b 8y 2 c y d 49x 2
a
b
c

twelve more than y
twelve added to y
y +12
d
seven less than 49x 2
seven subtracted from 49x 2
49x 2 −7
a 14x 2 b 12x c z
d 8x
a 17y 2 b 7 c d
a b
a (x + y).
eight times the sum of x and y
8(x + y)
b
a
b
a
b

2 3 −12÷(9−5) 3 2 −18÷(11−5) 2+8(6+1)
4+6(3+6) 20 ÷ 4 + 6(5−1) 33 ÷ 3 + 4(7−2)
3(1+9·6)−4 2 5(2 + 8 · 4) − 7 2 2 ⎡ ⎣1+3(10 − 2) ⎤ ⎦
5 ⎡ ⎣2+4(3−2) ⎤ ⎦
8+2 ⎡ ⎣7−2(5−3) ⎤ ⎦ −3 2 10 + 3 ⎡ ⎣6−2(4−2) ⎤ ⎦ −2 4
x =2,
x =3,
a x 6
b 4 x
c 2x 2 +3x −7
a x 5
b 5 x
c 3x 2 −4x −8
x =4,y =1
x 2 +3xy −7y 2
x =3,y =2
x = 10, y =7
a =3,b =8
6x 2 +3xy −9y 2 (x − y) 2 a 2 + b 2
7x +2+3x +4
8y +5+2y −4
10a +7+5a −2+7a −4
7c +4+6c −3+9c −1

3x 2 +12x +11+14x 2 +8x +5
5b 2 +9b +10+2b 2 +3b −4
a 5x 2 6xy
b 6y 2 5x
c y 2
d 6x 81x 2
a 4ab 2 3a 2 b
b 4y 2 5x
c m
d 9x 121x 2
a y
b y
a 17x 2 5xy
b 8y 3 3x
c a 2
d 11b 100b 2
a 3x 2 y 7xy 2
b 6xy 2 4z
c 3x 2
d 7x 2 63x 3
a y
b y
a 3x y
b 3x y
a 4x 2 5x
b 4x 2 5x
m
$5 $10
t
8a 2 +4a +9−a 2 −1.

a
b

2 −2
2 −2, −2 2.
−3
−a means the opposite of the number a
The notation −a is read as “the opposite of a.”
−3

n |n| |n| ≥0
−5 is 5 units away from 0, so |−5| =5.
5 is 5 units away from 0, so |5| =5.
−5
, =
a |−5|__ − |−5| b 8__ − |−8| c −9__ − |−9| d −(−16)__|−16|.
a
|−5|__ − |−5|
5__ − 5
5> −5
|−5| >−|−5|
b
8__ − |−8|
8__ − 8
8> −8
8> −|−8|
c

−9__ − |−9|
−9__−9
−9 = −9
−9 = − |−9|
d
−(−16)__|−16|
16__16
16 = 16
−(−16) = |−16|
< , > , =
a −9__ − |−9| b 2__ − |−2| c −8__|−8| d −(−9)__|−9|.
< , > , =
a 7__ − |−7| b −(−10)__|−10| c |−4|__ − |−4| d −1__|−1|.
Parentheses () Braces {}
Brackets [] Absolute value ||
24 − |19 − 3(6−2)|.

24 − |19 − 3(6−2)|
24 − |19 − 3(4)|
24 − |19 − 12|
24 − |7|
24 − 7
17
19 − |11 − 4(3 − 1)|.
9 − |8 − 4(7 − 5)|.
Counting numbers
Whole numbers
1, 2, 3 …
0, 1, 2, 3….
… −3, −2, −1, 0, 1, 2, 3…
… −3, −2, −1, 0, 1, 2, 3…,
5+3 −5+(−3) −5+3 5+(−3)
5+3,

−5 + (−3),
−5+3 5+(−3).
a −1 + (−4) b −1 + 5 c 1+(−5).
a
b
c

a −2 + (−4) b −2 + 4 c 2 + (−4).
a −2 + (−5) b −2 + 5 c 2 + (−5).
“5 − 3”
5−3 −5−(−3) −5−3 5−(−3)
5−3,
−5 − (−3),
−5−3 5−(−3).

a 3−1 b −3 − (−1) c −3−1 d 3−(−1).
a
b
c
d
a 6 − 4 b −6 − (−4) c −6 − 4 d 6 − (−4).
a 7 − 4 b −7 − (−4) c −7 − 4 d 7 − (−4).

−3 − 1
−3 + (−1) 3−(−1) 3+1.
a − b = a + (−b)
a 13 − 8 13 + (−8) b −17−9 −17 + (−9) c 9 − (−15) 9+15 d −7 − (−4)
−7+4.
a
13 − 8 and 13 + (−8)
5 5
b
−17 − 9 and −17 + (−9)
−26
−26
c
9−(−15) and 9 + 15
24 24
d
−7 − (−4) and −7 + 4
−3
−3
a 21 − 13 21 + (−13) b −11 − 7 −11 + (−7) c 6 − (−13) 6 + 13 d −5 − (−1)
−5 + 1.
a 15 − 7 15 + (−7) b −14 − 8 −14 + (−8) c 4 − (−19) 4 + 19 d −4 − (−7)
−4 + 7.

7−(−4−3) −9.
7−(−4 − 3) −9
7−(−7) −9
14 − 9
5
8 − (−3 − 1) − 9.
12 − (−9 − 6) − 14.
a · b
−3? 5, 3

5·3=15 −5(3) = −15
5(−3) = −15 (−5)(−3) =15
signs are the same, the product is positive.
signs are different, the product is negative.
15 ÷ 3 = 5 5 · 3 = 15.
5·3=15 so 15÷3=5 −5(3) = −15 so −15 ÷ 3 = −5
(−5)(−3) =15 so 15÷(−3) =−5 5(−3) = −15 so −15 ÷ (−3) =5

a −100 ÷ (−4) b 7·6 c 4(−8) d −27 ÷ 3.
a
−100 ÷ (−4)
25
b
7·6
42
c
4(−8)
−32
d
−27÷3
−9
a −115 ÷ (−5) b 5 · 12 c 9(−7) d −63 ÷ 7.
a −117 ÷ (−3) b 3 · 13 c 7(−4) d −42 ÷ 6.
−1,
−1
−1a =−a
−1

a (−2) 4 b −2 4 .
−2
a
(−2) 4
(−2)(−2)(−2)(−2)
4(−2)(−2)
−8(−2)
16
b
−2 4
−(2·2·2·2)
2 4
−(4·2·2)
−(8·2)
−16
a (−3) 4 b −3 4 .
a (−7) 2 b −7 2 .
(−2) 4 −2 4 .
a 8(−9) ÷ (−2) 3 b −30 ÷ 2 + (−3)(−7).
a

8(−9) ÷ (−2) 3
8(−9) ÷ (−8)
−72 ÷ (−8)
9
b
−30÷2+(−3)(−7)
−15 + (−3)(−7)
−15+21
6
a 12(−9) ÷ (−3) 3 b −27 ÷ 3 + (−5)(−6).
a 18(−4) ÷ (−2) 3 b −32 ÷ 4 + (−2)(−7).
4x 2 −2xy +3y 2 x =2,y =−1.
3x 2 − 2xyx + 6y 2 x = 1, y = −2.
4x 2 − xy
+ 5y 2 x = −2, y = 3.

−12, 3.
the sum of – 8 ––– and −12 increased by 3
⎡
⎣8+(−12) ⎤ ⎦ +3
(−4) +3
−1
−16,
−8 −12,
−9

15
−6 −15

, =
a |−7|___ − |−7|
b 6___ − |−6|
c |−11|___ − 11
d −(−13)___ − |−13|
a −|−9|___|−9|
b −8___|−8|
c |−1|___ − 1
d −(−14)___ − |−14|
a −|2|___ − |−2|
b −12___ − |−12|
c |−3|___ − 3
d |−19|___ − (−19)
a −|−4|___ − |4|
b 5___ − |−5|
c −|−10|___ − 10
d −|−0|___ − (−0)
|15 − 7| − |14 − 6| |17 − 8| − |13 − 4| 18 − |2(8−3)|
15 − |3(8−5)| 18 − |12 − 4(4−1) +3| 27 − |19 + 4(3−1) −7|
10 − 3|9−3(3−1)|
13 − 2|11 − 2(5−2)|
a −7 + (−4)
b −7+4
c 7+(−4).
a −5 + (−9)
b −5+9
c 5+(−9)
48 + (−16)
34 + (−19) −14 + (−12) +4 −17 + (−18) +6
19 + 2(−3 + 8) 24 + 3(−5 + 9)
a 13 − 7
b −13 − (−7)
c −13−7
d 13 − (−7)

a 15 − 8
b −15 − (−8)
c −15−8
d 15 − (−8)
−17 − 42
−58 − (−67)
−14 − (−27) +9 64 + (−17) −9 a 44 − 28 b 44 + (−28)
a 35 − 16 b 35 + (−16) a 27 − (−18) b 27 + 18 a 46 − (−37) b 46 + 37
(2−7) − (3−8) (1−8) − (2−9) −(6−8) − (2−4)
−(4−5) − (7−8) 25 − ⎡ ⎣10 − (3−12) ⎤ ⎦ 32 − ⎡ ⎣5−(15 − 20) ⎤ ⎦
a −4 · 8
b 13(−5)
c −24 ÷ 6
d −52 ÷ (−4)
a −3 · 9
b 9(−7)
c 35 ÷ (−7)
d −84 ÷ (−6)
a −28 ÷ 7
b −180 ÷ 15
c 3(−13)
d −1(−14)
a −36 ÷ 4
b −192 ÷ 12
c 9(−7)
d −1(−19)
a (−2) 6 b −2 6 a (−3) 5 5
b −3 5(−6) +7(−2) −3
8(−4) +5(−4) −6 −3(−5)(6) −4(−6)(3)
(8−11)(9−12) (6−11)(8−13) 26 − 3(2−7)
23 − 2(4−6) 65 ÷ (−5) + (−28) ÷ (−7) 52 ÷ (−4) + (−32) ÷ (−8)
9−2 ⎡ ⎣3−8(−2) ⎤ ⎦ 11 − 3 ⎡ ⎣7−4(−2) ⎤ ⎦ 8−|2−4(4−1) +3|
7−|5−3(4−1) −6| 9−3|2(2−6) − (3−7)| 5−2|2(1−4) − (2−5)|
(−3) 2 −24÷(8−2) (−4) 2 −32÷(12 − 4)

y + (−14)
a y = −33 b y =30
x + (−21)
a x = −27 b x =44
(x + y) 2
x =−3,y =14
(y + z) 2
y =−3,z =15
9a −2b −8
a =−6 b =−3
7m −4n −2
m =−4 n =−9
3x 2 −4xy +2y 2
x =−2,y =−3
4x 2 − xy +3y 2
x =−3,y =−2
−15, −8 −9,
23
a 10 −18
b 11 −25
a −5 −30
b −6 −13
−6
a b
−13
c d
−12°.
89°,
−31°.
−$49
−8
−2
−4 3 = (−4) 3 ?

a
b

2 3
2 ,
3
2
3
a , b ≠0
b
b
a
b ≠0,c ≠0,
a b = a · c
b · c a · c
b · c = a b .
2 3.
3
10 15.
15

− 315
770 .
− 69
120 .
− 120
192 .
b ≠0, d ≠0,
a · = ac
b
cd bd
a 1 . 3=3 1 .

− 12 5 (−20x).
11 3 (−9a).
13 7 (−14b).
2 3 3 2 . 4 ,
1
1 4 .
b ≠0,c ≠0, d ≠0,
a
b ÷ d c = a b · dc
b ≠0,
c ≠0, d ≠0,
−
18 7 ÷ ⎛ ⎝ −14 ⎞
27⎠ .

−
27 7 ÷ ⎛ ⎝ − 35
⎞
36⎠
.
−
14 5 ÷ ⎛ ⎝ − 15
⎞
28⎠
.
6
7
3
3 4 ÷ 5 8 .
3
4
5
8
x
2
5
6
3
4
5
8
x
2
xy.
6

x
2
xy
6
x
2 ÷ xy
6
x
2 · 6xy
x ·6
2·xy
x ·3·2
2 · x · y
3 y
a
8 .
ab
6
p
2
pq .
8
c ≠0,
a c + b c = a + b c
and a c − b c = a − b c
7
12
+ 5 18 .

7
12
+ 11
15 .
13
15 + 17
20 .

a · = ac
b
cd bd
a c + b c = a + b c
a
b ÷ d c = a b · dc
a c − b c = a − b c
a 5x
6 − 3 10 b 5x
6 · 310 .
a
5x
6
− 3 10
6=2·3
___________ 10=2·5
LCD=2·3·5
LCD = 30
5x ·5
6·5 − 3·3
10 · 3
25x
30 − 9 30
25x −9
30
b

5x
6 · 310
5x ·3
6·10
5 x ·3
2·3·2·5
x
4
5x
6 − 10 3 , 5x
6 · 310 .
a 3a 4 − 8 9 b 3a 4 · 89 .
a 4k
5 − 1 6 b 4k
5 · 16 .
−1
3
=− 1 3
1
−3 =−1 3
negative
positive = negative
positive
negative = negative
−a
b
= a −b =−a b
4(−3) +6(−2)
.
−3(2) −2

4(−3) +6(−2)
−3(2) −2
−12 + (−12)
−6 − 2
−24
−8
3
8(−2) + 4(−3)
.
−5(2) + 3
7(−1) + 9(−3)
.
−5(3) − 2
⎛
⎝ 1 2
⎞
2⎠
4+3 2.
⎛1
⎛ ⎞ 2
⎝3
⎠
2 3 + 2 .

1 + 42
⎛1
⎛ ⎞ 2 .
⎝4
⎠
1
2 + 2 3
3
4 − 1 .
6
⎛
⎝ 1 2 + 2 ⎞
3⎠
⎛
⎝ 3 4 − 1 ⎞
6⎠
⎛
⎝ 3 6 + 4 ⎞
6⎠
⎛
⎝ 9 12 − 2 ⎞
12⎠
⎛
⎝ 7 ⎞
6⎠
⎛
⎝ 7 ⎞
12⎠
7
6 ÷ 7 12
7
6 ⋅ 12 7
7 ⋅6⋅2
6 ⋅7⋅1
2
1
3 + 1 2
3
4 − 1 .
3
2
3 − 1 2
1
4 + 1 .
3

2x 2 y x = 1 4 y =−2 3 .
2·2·4
3ab 2 a = − 2 3 b = − 1 2 .
4c 3 d c = − 1 2 d = − 4 3 .

− 108
63
− 104
48
120
252
182
294 14x 2
21y
24a
32b 2
− 210a2
110b 2
− 30x2
105y 2
− 3 ⎛
4⎝ −4 ⎞
9⎠
− 3 8 · 415
⎛ ⎝ −14 ⎞
15⎠ ⎛ ⎝20
9 ⎞
⎠
⎛ ⎝ − 10
9 ⎞
⎠ ⎛ ⎝ 25 ⎞
33⎠
⎛ ⎝ −63 ⎞ ⎛
84⎠⎝ −44 ⎞
90⎠
⎛ ⎝ −33 ⎞
60⎠ ⎛ ⎝ −40 ⎞
88⎠
3
7 ·21n 5
6 ·30m 3
4
÷ x
11
2
5
÷ y 9
5 ÷ ⎛ 18 ⎝ −15 ⎞
24⎠ 7 ÷ ⎛ 18 ⎝ −14 ⎞
27⎠
8u
15 ÷ 12v
25
−15 ÷ ⎛ ⎝ −5 ⎞
3⎠
12r
25 ÷ 18s
35
3
4
÷ (−12)
− 8 21
12
35
− 9 16
33
40
− 4 5
2
5
3
10
m
3
n
2
− 3 8
− y
12
7
12
+ 5 8
5
12
+ 3 8
7
12
− 9 16
7
16
− 5 12
− 13
30 + 25
42
− 23
30 + 5 48

− 39
56 − 22
35
− 3 4 − ⎛ ⎝ −4 ⎞
5⎠
− 33
49 − 18
35
x
3
+ 1 4
− 2 3 − ⎛ ⎝ −3 ⎞
4⎠
x
5
− 1 4
a 2 3 + 1 6
b 2 3 ÷ 1 6
a − 2 5 − 1 8
b − 2 5 · 18
a 5n 6 ÷ 8 15
b 5n 6 − 8 15
a 3a 8 ÷ 7 12
b 3a 8 − 7 12
a − 5a 3 + ⎛ ⎝ −10 ⎞
6 ⎠
b − 5a 3 ÷ ⎛ ⎝ −10 ⎞
6 ⎠
a − 4x
9 − 5 6
b − 4k
9 · 56
a 2b 5 + 8 15
b 2b 5 ÷ 8 15
a − 3y
8 − 4 3
b − 3y
8 · 43
5·6−3·4
4·5−2·3
8·9−7·6
5·6−9·2 5 2 −3 2
3−5
6 2 −4 2
4−6
7·4−2(8−5)
9·3−3·5
9·7−3(12 − 8)
8·7−6·6
9(8−2) −3(15 − 7)
6(7−1) −3(17 − 9)
8(9−2) −4(14 − 9)
7(8−3) −3(16 − 9)
2 3 +4 2
2
⎛
⎝ 2 ⎞
3⎠
3 3 −3 2
2
⎛
⎝ 3 ⎞
4⎠
⎛
⎝ 3 2
⎞
5⎠
⎛
⎝ 3 2
⎞
7⎠
⎛
⎝ 3 2
⎞
4⎠
⎛
⎝ 5 2
⎞
8⎠
2
1
3 + 1 5
5
1
4 + 1 3
7
8 − 2 3
1
2 + 3 8
3
4 − 3 5
1
4 + 2 5
− 3 8 ÷ ⎛ ⎝ − 10
3 ⎞
⎠ −
12 3 ÷ ⎛ ⎝ −5 ⎞
9⎠
− 3 8 + 5 12

− 1 8 + 7 12
− 7 15 − y 4
− 3 8 − x
11
11
12a · 9a
16
10y
13 · 8
15y
1
2
+ 2 3 · 512
1
3
+ 2 5 · 34
1− 3 5 ÷ 1 10
1− 5 6 ÷ 1 12
3
8
− 1 6 + 3 4
2
5
+ 5 8 − 3 4
12 ⎛ ⎝20 9 − 15
4 ⎞
⎠
8 ⎛ ⎝ 15
16 − 5 ⎞
6⎠
5
8 + 1 6
19
24
1
6 + 3 10
14
30
⎛ ⎝ 5 9 + 1 ⎞
6⎠ ÷ ⎛ ⎝ 2 3 − 1 ⎞
2⎠ ⎛ ⎝ 3 4 + 1 ⎞
6⎠ ÷ ⎛ ⎝ 5 8 − 1 ⎞
3⎠
7
10
− w
5
12
− w
2x 2 y 3
a w = 1 2 b w =−1 2
8u 2 v 3
u =− 3 4 v =−1 2
a w = 1 4 b w =−1 4
a + b
a − b
a =−3,b =8
x =− 2 3 y =−1 2
r − s
r + s
r = 10, s =−5
a
b

0.1 =
10
1
0.01 =
100
1
0.001 = 1
1000
0.0001 = 1
10,000
is “one tenth”
is “one hundredth”
is “one thousandth”
is “one ten-thousandth”

a b c
18.379.
a
b

c
6.582 a b c
15.2175 a b c
a −23.5 − 41.38 b 14.65 − 20.
a

−23.5 − 41.38
23.5
______ +41.38
5 = 50
23.50
10 100 0.5 = 0.50 ______ +41.38
23.50
______ +41.38
64.88
−23.5 − 41.38 = −64.88
b
14.65 − 20
20
______ −14.65
20.00
______ −14.65
9 9
1 10 10 10
2 0 . 0 0
______________
−1 4 . 6 5
5. 3 5
14.65 − 20 = −5.35
a −4.8 − 11.69 b 9.58 − 10.
a −5.123 − 18.47 b 37.42 − 50.

(−3.9)(4.075).
(−3.9)(4.075)
(−3.9)(4.075) = −15.8925
−4.5(6.107).
−10.79(8.12).

a b c
a
b
c
a b c
a b c

−25.65 ÷ (−0.06).
−23.492 ÷ (−0.04).
−4.11 ÷ (−0.12).
0.03.
0.03 = 3
100

a 0.374 b − 5 8
a
b 5 8
85.
a 0.234 b − 7 8
a 0.024 b − 3 8

a
b
a
b
a
b
a
b

n n 2 n
8 2
11 2
n 2 = m,
(−3) 2 =9 (−8) 2 =64 (−11) 2 = 121 (−15) 2 = 225
10 2 = 100,
n 2 = m,
(−10) 2 = 100 −10 −10
m
m m
m = n 2 , m = n, n ≥0.
m,
100 = 10.
− 100 = −10. − 100
a 25 b 121 c − 144.
a
5 2 =25 5
25
b

11 2 = 121 11
121
c
− 144
−12
a 36 b 169 c − 225.
a 16 b 196 c − 100.
Counting numbers 1, 2, 3, 4, …..
Whole numbers 0, 1, 2, 3, 4, ….
Integers
…. − 3, −2, −1, 0, 1, 2, 3, ….
−1.2684
1 3 0. 3– .
q p , q ≠0,
q p , q ≠0.
π
π = 3.141592654...

−25? −25?
() 2 = −25?
−25.
−25.
,8, 5, 5.9, − 64, a b c d
−7, 14 5
e
a 0, 1, 2, 3, …,

−7 −64=−8.
−7, 8, −64.
c −7, 8, −64
14 5 5.9 −7, 14 , 8, 5.9, −64.
5
d
5
e
−3, − 2, 0. 3 – , 9 5 , 4,
49, a b c
d e
− 25, − 3 , −1, 6, 121, 2.041975..., a b c
8
d e
1
5 , − 4 5 ,3,7 4 , − 9 2 ,−5 8 3
−5
1 5 −4 5 . 1 5
1 5 , 2 5 , 3 5 , 4 .
5
1
5 .
− 4 5 −1. −4 5 .
7 4 , 9 2 , 8 .
3
7
4 =13 −
4
9 2 =−41 2
8
3 =22 3
4, 3 4 , − 1 4 ,−3,6 5 , − 5 2 , 7 3 .

4, −3.
3 4 3 4
3 4 . −1 4 .
6 5 , − 5 2 ,
7 .
3
6 5 =11 5 , − 5 2 =−21 2 , 7 3 =21 3 .
−1, 1 3 , 6 5 , − 7 4 , 9 2 , 5, − 8 3 .
−2, 2 3 , 7 5 , − 7 4 , 7 2 , 3, − 7 3 .
a b −0.74.
a
10 4 ,
b −0.74 −
100 74 , −1.
−1.
a 0.6 b −0.25.
a 0.9 b −0.75.

a b c
−16.53 − 24.38 −19.47 − 32.58 −38.69 + 31.47
−29.83 + 19.76 72.5 − 100 86.2 − 100
91.75 − (−10.462) 94.69 − (−12.678) 55.01 − 3.7
59.08 − 4.6 2.51 − 7.4 3.84 − 6.1
⎛ ⎝94.69)(−12.678) (−8.5)(1.69) (−5.18)(−65.23)
(−9.16)(−68.34) (0.06)(21.75) (0.08)(52.45)
(9.24)(10) (6.531)(10) (0.025)(100)
(0.037)(100) (55.2)(1000) (99.4)(1000)
$117.25 ÷ 48 $109.24 ÷ 36 1.44 ÷ (−0.3)
−1.15 ÷ (−0.05) 5.2 ÷ 2.5 14 ÷ 0.35
0.04 0.095
−0.375
17
20
− 18
11
17
4
− 310
25

71% 150% 39.3%
7.8%
1.56 0.0625
2.254
64 169 144
−4 − 100 − 121
a b c d e
−8, 0, 1.95286..., 12 5 , 36,9 −9, −3 4 —
,− 9, 0.4 09 , 11,7
9 6
− 100, −7, − 8 3 , −1, 0.77, 31 4
−6, − 5 ———
, 0, 0.714285 ,2 1, 14
2 5
3
10
, 7 2 , 11 6 ,4 7
10
, 5 2 , 13 8 ,3 3
4
, − 3 4 ,12 3 ,−12 3 , 5 2 , − 5 2
2
5
, − 2 5 ,13 4 ,−13 4 , 8 3 , − 8 3
a 0.8 b −1.25 a −0.9 b −2.75
a −1.6 b 3.25 a 3.1 b −3.65

a
b

8+9 9+8,
8+9=9+8.
9·8 8·9
9·8=8·9.
of Addition If a and b are real numbers, then a + b = b + a.
of Multiplication If a and b are real numbers, then a · b = b · a.
9−8 8−9 9−8≠8−9.
12 ÷ 3 ≠ 3 ÷ 12,
(7+8) +2=7+(8+2),
⎛
⎝ 5·13 ⎞
⎠·3=5·⎛⎝ 1 3 ·3⎞ ⎠ ,
of Addition If a, b, and c are real numbers, then (a + b) + c = a + (b + c).
of Multiplication If a, b, and c are real numbers, then (a · b) · c = a · (b · c).

(10 − 3) −2≠10−(3−2) (24 ÷ 4) ÷2≠24÷(4 ÷2)
7−2≠10−1
6÷2≠24÷2
5≠9
3≠12
18p +6q +15p +5q.
18p +6q +15p +5q
18p +15p +6q +5q
33p +11q
23r + 14s + 9r +15s.
37m + 21n + 4m − 15n.
⎛ ⎝13 5 + 3 ⎞
4⎠ + 1 4 .
⎛
⎝13 5 + 3 ⎞
4⎠ + 1 4
5
13 + ⎛ ⎝ 3 4 + 1 ⎞
4⎠
5
13 + ⎛ ⎝ 4 ⎞
4⎠
5
13 +1
1 5 13
18
13
⎛ 7
⎝ + 5 ⎞
15 8⎠
+ 3.
8

⎛2 ⎝ + 7 ⎞
9 12⎠
+ 5 . 12
a, a +0=a 0+a = a.
a, a ·1=a
1·a = a.
of Addition For any real number a : a +0=a 0+a = a
0 is the additive identity
of Multiplication For any real number a : a ·1=a 1·a = a
1 is the multiplicative identity
−a a.
a, a + (−a) =0.
2 3 2
3
1 a
a, a ≠0,a · 1a =1.
of Addition For any real number a, a + (−a) =0
−a is the additive inverse of a
A number and its opposite add to zero.
of Multiplication For any real number a, a ≠0, a · 1a =1.
1
a is the multiplicative inverse of a.
A number and its reciprocal multiply to one.
0÷3?
0÷3=0.
0÷3=0 0·3=0.
0?

4?
4÷0
a ·0=0 0·a = 0 The product of any number and 0 is 0.
a ≠0
a
0
0
a = 0
is undefined Division by zero is undefined.
Zero divided by any real number, except itself, is zero.
−84n + (−73n) +84n.
−84n + (−73n) +84n
−84n +84n + (−73n)
0+(−73n)
−73n
−27a + (−48a) + 27a.
39x + (−92x) + (−39x).
7
15 · 823 · 15 7 .

7
15 · 823 · 15 7
7
15 · 15 7 · 823
1· 823
8
23
9
16 · 549 · 16 9 .
6
17 · 11
25 · 17 6 .
a
n +5 0
10 − 3p
, n ≠−5b , 10 − 3p ≠0.
0
a
0
n +5
Zero divided by any real number except itself is 0. 0
b
Division by 0 is undefined.
10 − 3p
0
undefined
a
m 0 , m ≠ −7 b 18 − 6c, 18 − 6c ≠ 0.
+ 7 0
a
d − 0
15 − 4q
, d ≠ 4 b , 15 − 4q ≠ 0.
4 0

If a, b, and c are real numbers, then
a(b + c) = ab + ac
(b + c)a = ba + ca
a(b − c) = ab − ac
(b − c)a = ba − ca
3(x +4).
3(x +4)
3·x +3·4
3x +12
4(x + 2).
6(x + 7).
8 ⎛ ⎝ 3 8 x + 1 ⎞
4⎠ .
6 ⎛5 ⎝ y + 1 ⎞
6 2⎞
⎠ .
12 ⎛1 ⎝ n + 3⎞ 3 4⎠ .

100 ⎛ ⎝0.3 + 0.25q ⎞ ⎠.
100 ⎛ ⎝ ⎛ 0.7 + 0.15p ⎞ ⎠ ⎞ .
100(0.04 + 0.35d).
−11(4−3a).
−11(4−3a)
−11 · 4 − (−11) ·3a
−44 − (−33a)
−44+33a
33a − 44.
−5(2 − 3a).
−7 ⎛ ⎝ ⎛ 8 − 15y ⎞ ⎠ ⎞ .
− ⎛ ⎝y +5 ⎞ ⎠.

− ⎛ ⎝y +5 ⎞ ⎠
−1
−1 ⎛ ⎝y +5 ⎞ ⎠
−1 · y + (−1) ·5
−y + (−5)
−y −5
−(z − 11).
−(x − 4).
8−2(x +3)
8−2(x +3)
8−2·x −2·3
8−2x −6
−2x +2
9 − 3(x + 2).
7x − 5(x + 4).
4(x −8) − (x +3).

4(x −8) − (x +3)
4x −32−x −3
3x −35
6(x − 9) − (x + 12).
8(x − 1) − (x + 5).
of addition If a, b are real numbers, then a + b = b + a
of multiplication If a, b are real numbers, then a · b = b · a
of addition If a, b, and c are real numbers, then (a + b) + c = a + (b + c)
of multiplication If a, b, and c are real numbers, then (a · b) · c = a · (b · c)
If a, b, and c are real numbers, then a(b + c) = ab + ac
(b + c)a = ba + ca
a(b − c) = ab − ac
(b − c)a = ba − ca
of addition For any real number a:
a +0=a
0 is the additive identity 0+a = a
of multiplication For any real number a:
a ·1=a
1 is the multiplicative identity 1·a = a

of addition For any real number a, a + (−a) =0
−a is the additive inverse of a
A number and its opposite add to zero.
of multiplication For any real number a, a ≠0 a · 1a =1
1
a is the multiplicative inverse of a
A number and its reciprocal multiply to one.
For any real number a, a ·0=0
0·a =0
For any real number a, a ≠0, 0 a =0
For any real number a,
a
0
is undefined

43m + (−12n) + (−16m) + (−9n)
−22p +17q + ⎛ ⎝−35p ⎞ ⎠ + ⎛ ⎝−27q ⎞ ⎠
3
8
g + 1 12 h + 7 8 g + 5 12 h
5
6
a + 3 10 b + 1 6 a + 9 10 b
6.8p + 9.14q + ⎛ ⎝−4.37p ⎞ ⎠ + ⎛ ⎝−0.88q ⎞ ⎠
−24 · 7 · 38
9.6m + 7.22n + (−2.19m) + (−0.65n)
−36·11·49
⎛ ⎝ 5 6 + 15
8 ⎞
⎠ + 15
7
⎛ ⎝ 11
12 + 4 ⎞
9⎠ + 5 9
17(0.25)(4) 36(0.2)(5)
⎡ ⎣2.48(12) ⎤ ⎦(0.5)
⎡ ⎣9.731(4) ⎤ ⎦(0.75)
12 ⎛ ⎝ 5 6 p⎞ ⎠
20 ⎛ ⎝ 3 5 q⎞ ⎠
19a +44−19a
27c +16−27c
1 + 7 2 8 + ⎛ ⎝ −1 ⎞
2⎠ 2 + 5 5 12 + ⎛ ⎝ −2 ⎞
5⎠
10(0.1d)
3
20 · 49
11 · 20 3
100 ⎛ ⎝0.01p ⎞ ⎠
13
18 · 25 7 · 18
13
0
u − 4.99 , u ≠ 4.99 0÷⎛ ⎝ y − 1 ⎞
6⎠ , x ≠ 1 6
32 − 5a
0
, 32 − 5a ≠0 28 − 9b, 28 − 9b ≠0
0
⎛ ⎝ 3 4 + 10 9 m⎞ ⎠ ÷0, 3 4 + 10 9 m ≠0 ⎛ ⎝16 5 n − 3 ⎞
7⎠ ÷0, 5
16 n − 3 7 ≠0
8 ⎛ ⎝4y +9 ⎞ ⎠ 9(3w +7) 6(c −13)

7 ⎛ ⎝y −13 ⎞ ⎠
1
4
⎛
⎝3q +12 ⎞ ⎠ 1
5
(4m +20)
9 ⎛ ⎝ 5 9 y − 1 ⎞
3⎠ 10 ⎛ ⎝10 3 x − 2 ⎞
5⎠
12 ⎛ ⎝ 1 4 + 2 3 r⎞ ⎠
12 ⎛ ⎝ 1 6 + 3 4 s⎞ ⎠
15 · (4d +10) 18 · (15h +24)
35
56
r(s −18) u(v −10) ⎛ ⎝y +4 ⎞ ⎠p
(a +7)x −7 ⎛ ⎝4p +1 ⎞ ⎠ −9(9a +4)
−3(x −6) −4 ⎛ ⎝q −7 ⎞ ⎠ −(3x −7)
− ⎛ ⎝5p −4 ⎞ ⎠ 16 − 3 ⎛ ⎝y +8 ⎞ ⎠ 18 − 4(x +2)
4−11(3c −2) 9−6(7n −5) 22 − (a +3)
8−(r −7) (5m −3) − (m +7) ⎛ ⎝4y −1 ⎞ ⎠ − ⎛ ⎝y −2 ⎞ ⎠
9(8x −3) − (−2) 4(6x −1) − (−8) 5(2n +9) +12(n −3)
9(5u +8) +2(u −6) 14(c −1) −8(c −6) 11(n −7) −5(n −1)
6 ⎛ ⎝7y +8 ⎞ ⎠ − ⎛ ⎝30y −15 ⎞ ⎠ 7(3n +9) − (4n −13)
8 ⎛ ⎝ x − 1 ⎞
4⎠
4($5.97)
$5.97 6 − 0.03
a
b

0
a ,
b
b ≠0,
a · b = m,
a , b ≠0,
b
0
a , b ≠0,
b
q p , q ≠0.

n 2 = m,
n 2 = m,
a = b

a ≠ b
a < b
a ≤ b
a > b
a ≥ b
Parentheses ()
Brackets []
Braces {}
a n
a n n th

a
b
a b
a + b
a − b
a · b, ab, a(b), (a)(b)
2a
a ÷ b, a/b, a b , ba
−a means the opposite of the number a
The notation −a is read as “the opposite of a.”
|n| |n| ≥0
a − b = a + (−b)
Parentheses () Braces {}
Brackets [] Absolute value ||

−1
−1a =−a
−1
b ≠0,c ≠0,
a
b = a b · · c
c and a b · · c
c = a b .
b ≠0, d ≠0,
a · = ac
b
cd bd .
b ≠0,c ≠0, d ≠0,
a
b ÷ c d = a b · dc .
c ≠0,
a c + b c = a + c
b and a c − b c = a − c b .

−a
b
= a −b =−a b .
5?

m
m = n 2 , m = n, n ≥0.
m,
of addition If a, b are real numbers, then a + b = b + a
of multiplication If a, b are real numbers, then a · b = b · a
of addition If a, b, and c are real numbers, then (a + b) + c = a + (b + c)
of multiplication If a, b, and c are real numbers, then (a · b) · c = a · (b · c)

If a, b, and c are real numbers, then a(b + c) = ab + ac
(b + c)a = ba + ca
a(b − c) = ab − ac
(b − c)a = ba − ca
of addition For any real number a:
a +0=a
0 is the additive identity 0+a = a
of multiplication For any real number a:
a ·1=a
1 is the multiplicative identity 1·a = a
of addition For any real number a, a + (−a) =0
−a is the additive inverse of a
A number and its opposite add to zero.
of multiplication For any real number a, a ≠0 a · 1a =1
1
a is the multiplicative inverse of a
A number and its reciprocal multiply to one.
For any real number a, a ·0=0
0·a =0
For any real number a, a ≠0, 0 a =0
For any real number a,
a
0
is undefined
24 ÷ 3 + 4(5−2) 7+3 ⎡ ⎣6−4(5−4) ⎤ ⎦ −3 2

x =4, a x 3 b 5 x c 2x 2 −5x +3 2x 2 −4xy −3y 2 x =3, y =1
12y +7+2y −5 14x 2 −9x +11−8x 2 +8x −6
a 4ab 2 7a 3 b 2
b 6y 2 3y
c 5x
d 5y 8y 2
a y
b y
n
, =
a −|7|___ − |−7|
b −8___ − |−8|
c |−13|___ − 13
d |−12|___ − (−12)
9−|3(4−8)| 12 − 3|1−4(4−2)|
−12 + (−8) +7
a 15 − 7
b −15 − (−7)
c −15−7
d 15 − (−7)
−11 − (−12) +5
a 23 − (−17) b 23 + 17 −(7−11) − (3−5)

a −27÷9 b 120 ÷ (−8)
c 4(−14) d −1(−17)
a (−7) 3 3
b −7 (7−11)(6−13) 63 ÷ (−9) + (−36) ÷ (−4)
6−3|4(1−2) − (7−5)| (−2) 4 −24÷(13 − 5)
(y + z) 2
y =−4,z =7
3x 2 −2xy +4y 2
x =−2,y =−3
−4 −9, a
−8 b −25
204
228 − 270x3
198y 2
⎛
⎝ −14 ⎞
15⎠ ⎛ ⎝ 10 ⎞
21⎠ 6x
25 ÷ 9y
20
− 4 9
8
21
5 + 7 11
18 12
36 − 15
48
a 5 8 + 3 4 b 5 8 ÷ 3 4

a − 3y
10 − 5 6 b − 3y
10 · 56
4·3−2·5
−6·3+2·3
4(7−3) −2(4−9)
−3(4+2) +7(3−6)
4 3 −4 2
2
⎛
⎝ 4 ⎞
5⎠
4x 2 y 2
a + b
x = 2 a − b
3 y =−3 4
a =−4, b =6
6.738
a b c
−23.67 + 29.84 54.3 − 100 79.38 − (−17.598)
(−2.8)(3.97) (−8.43)(−57.91) (53.48)(10)
(0.563)(100) $118.35 ÷ 2.6 1.84 ÷ (−0.8)
0.65 −9.6
− 5 14
8
11
2.43 0.0475
289 −121
a b c d e

−8, 0, 1.95286..., 12 5 , 36,9
3
4
, − 3 4 ,11 3 ,−12 3 , 7 2 , − 5 2
a 3.2 b −1.35
5x + 5 8 12 y + 1 8 x + 12 7 y −32 · 9 · 58
⎛
⎝ 11
15 + 3 ⎞
8⎠ + 5 8
4 + 8 7 15 + ⎛ ⎝ −4 ⎞
7⎠
13 · · 15
15
917 13
0
x −3 , x ≠3
5x −7,5x −7≠0
0
8(a −4)
12 ⎛ ⎝ 2 3 b + 5 ⎞
6⎠
18 · (2x −5) 56
(x −5)p −4 ⎛ ⎝y −3 ⎞ ⎠ 12 − 6(x +3)
6(3x −4) − (−5) 5 ⎛ ⎝2y +3 ⎞ ⎠ − ⎛ ⎝4y −1 ⎞ ⎠

756.
5n +8+2n −1
x =−2
y =3: |3x −4y|
6
a
b −8 −3
n
28.1458
a b
a
11 5 b 1.15
3 5 , 2.8, and − 5 2
8+3 ⎡ ⎣6−3(5−2) ⎤ ⎦ −4 2 −(4−9) − (9−5) 56 ÷ (−8) + (−27) ÷ (−3)
16 − 2|3(1−4) − (8−5)| −5 + 2(−3) 2 −9 180
204
− 7 18 + 5 12
4 ÷ ⎛ 5 ⎝ −12 ⎞
25⎠
9−3·9
15 − 9
4 ⎛ ⎝−3+2(3−6) ⎞ ⎠
3 ⎛ ⎝11 − 3(2+3) ⎞ ⎠
5
13 ·47·13 5
− 5 9
10
21
−4.8 + (−6.7) 34.6 − 100 −12.04 · (4.2)
−8 ÷ 0.05 −121
⎛
⎝13 8 + 5 ⎞
7⎠ + 2 7
5x + ⎛ ⎝−8y ⎞ ⎠ −6x +3y a 0 9 b 11 0
−3(8x −5)
6 ⎛ ⎝3y −1 ⎞ ⎠ − ⎛ ⎝5y −3 ⎞ ⎠

3 (12x + 20).
2

5 − 2(n + 1).
5 6 1 4 .
5y +3=10y −4.
a y = 3 5 b y = 7 5
a
y = 3 5 y = 3 5
5y +3=10y −4.
b

y = 7 5 y = 7 5
5y +3=10y −4.
9y + 2 = 6y + 3.
a y = 4 3 b y = 1 3
4x − 2 = 2x + 1.
a x = 3 2 b x = − 1 2
a ≠0,
ax + b =0
7(n −3) − 8 = −15

2(m − 4) + 3 = −1.
5(a − 3) + 5 = −10.
2 (3m −6)=5−m.
3

m =3.
1 (6u + 3) = 7 − u.
3
2 (9x − 12) = 8 + 2x.
3
4(x −1) −2=5(2x +3) +6.
4x
10 > 4.
x =− 9 2 .

6 ⎛
⎝p
− 3 ⎞ ⎠ − 7 = 5 ⎛ ⎝4p + 3 ⎞ ⎠ − 12.
8 ⎛ ⎝ ⎛ q + 1 ⎞ ⎠ ⎞ − 5 = 3 ⎛ ⎝ ⎛ 2q − 4 ⎞ ⎠ ⎞ − 1.
10 ⎡ ⎣3−8(2s −5) ⎤ ⎦ =15(40 − 5s).
160s
s =−2.
6 ⎡ ⎣ ⎡ 4 − 2 ⎛ ⎝ ⎛ 7y − 1 ⎞ ⎠ ⎞⎤ ⎦⎤ = 8 ⎛ ⎝ ⎛ 13 − 8y ⎞ ⎠ ⎞ .
12 ⎡ ⎣ ⎡ 1 − 5(4z − 1) ⎤ ⎦ ⎤ = 3(24 + 11z).
7x + 8 = −13
−3,

7y +14=7 ⎛ ⎝y +2 ⎞ ⎠.
7y y’s
14 = 14
7y +14=7 ⎛ ⎝y +2 ⎞ ⎠
−8z =−8z +9?
8z
z’s
0≠9.
−8z =−8z +9 0=9. −8z =−8z +9
6(2n −1) +3=2n −8+5(2n +1).

12n
4 + 9(3x − 7) = −42x − 13 + 23(3x − 2).
8(1 − 3x) + 15(2x + 7) = 2(x + 50) + 4(x + 3) + 1.
8+3(a −4)=0.
a = 4 3 .
a = 4 3 .
11 ⎛ ⎝ ⎛ q + 3 ⎞ ⎠ ⎞ − 5 = 19.

6+14(k −8) = 95.
5m +3(9+3m) =2(7m −11).
14m
27 ≠ −22.
12c + 5(5 + 3c) = 3(9c − 4).
4(7d + 18) = 13(3d − 2) − 11d.

1
12
x + 5 6 = 3 4 .
1 4 x + 1 2 = 5 8 .
1 8 x + 1 2 = 1 4 .
5= 1 2 y + 2 3 y − 3 4 y.

y = 12.
7 = 1 2 x + 3 4 x − 2 3 x.
−1 = 1 2 u + 1 4 u − 2 3 u.
1 2 ⎛ ⎝y −5 ⎞ ⎠ = 1 4 ⎛ ⎝y −1 ⎞ ⎠.

y =9.
1 5 (n + 3) = 1 (n + 2).
4
1 2 (m − 3) = 1 (m − 7).
4
4q +3
+6=
2
3q +5
4

q =−5.
3r + 5
6
+1= 4r 3
+ 3 .
2s + 3
2
+ 1 = 3s 4
+ 2 .
0.7 =
10 7 0.29 = 29 .
100
0.25x + 0.05(x + 3) = 2.85.
0.25 =
100 25 , 0.05 = 100 5 , 2.85 = 2 85
100 .

x =9
0.25n + 0.05(n + 5) = 2.95.
0.10d + 0.05(d − 5) = 2.15.

6y +10=12y
4x +9=8x
a y = 5 3 b y =−1 2
a x =− 7 8 b x = 9 4
8u −1=6u
a u =− 1 2 b u = 1 2
9v −2=3v
a v =− 1 3 b v = 1 3
15 ⎛ ⎝y −9 ⎞ ⎠ = −60 −16(3n +4) =32
−(w −12) =30 −(t −19) =28
51 + 5(4 − q) =56 −6+6(5−k) =15
3(10 − 2x)+54=0
−2(11 − 7x)+54=4
2
3
(9c −3) =22 3
5
(10x −5) =27
1
5
(15c + 10) = c +7 1
4
(20d + 12) = d +7
3(4n −1)−2=8n +3 9(2m −3)−8=4m +7
12+2(5−3y) =−9 ⎛ ⎝y −1 ⎞ ⎠ −2 −15 + 4(2 − 5y) =−7 ⎛ ⎝y −4 ⎞ ⎠ +4
5 + 6(3s −5)=−3+2(8s −1) −12+8(x −5)=−4+3(5x −2)
4 ⎛ ⎝p −4 ⎞ ⎠ − ⎛ ⎝p +7 ⎞ ⎠ =5 ⎛ ⎝p −3 ⎞ ⎠ 3(a −2) − (a +6) =4(a −1)
4 ⎡ ⎣5−8(4c −3) ⎤ ⎦ =12(1−13c) −8 5 ⎡ ⎣9−2(6d −1) ⎤ ⎦ =11(4−10d) − 139
3 ⎡ ⎣−9 + 8(4h −3) ⎤ ⎦ =2(5−12h) −19
3 ⎡ ⎣−14 + 2(15k −6) ⎤ ⎦ =8(3−5k) −24
5 ⎡ ⎣2(m +4) +8(m −7) ⎤ ⎦ =2 ⎡ ⎣3(5+m) − (21 − 3m) ⎤ ⎦
10 ⎡ ⎣5(n +1) +4(n −1) ⎤ ⎦ =11 ⎡ ⎣7(5+n) − (25 − 3n) ⎤ ⎦
23z +19=3(5z −9) +8z +46 15y +32=2 ⎛ ⎝10y −7 ⎞ ⎠ −5y +46

18(5j − 1) + 29 = 47 24(3d − 4) + 100 = 52
22(3m −4)=8(2m +9) 30(2n −1)=5(10n +8)
7v +42=11(3v +8) −2(13v −1) 18u −51=9(4u +5) −6(3u −10)
45(3y −2)=9 ⎛ ⎝15y −6 ⎞ ⎠ 60(2x −1)=15(8x +5)
9(14d +9)+4d =13(10d +6) +3 11(8c +5)−8c =2(40c +25) +5
1
4
x − 1 2 =−3 4
3
4
x − 1 2 = 1 4
5
6
y − 2 3 =−3 2
5
6
y − 1 3 =−7 6
1
2
a + 3 8 = 3 4
5
8
b + 1 2 =−3 4
2= 1 3 x − 1 2 x + 2 3 x
2=3 5 x − 1 3 x + 2 5 x
1
3
w + 5 4 = w − 1 4
1
2
a − 1 4 = 1 6 a + 1 12
1
3
b + 1 5 = 2 5 b − 3 5
1
3
x + 2 5 = 1 5 x − 2 5
1
4
⎛
⎝p −7 ⎞ ⎠ = 1 3 ⎛ ⎝p +5 ⎞ ⎠
1
5
⎛
⎝q +3 ⎞ ⎠ = 1 2 ⎛ ⎝q −3 ⎞ ⎠
1
2
(x +4) = 3 4
1
3
(x +5) = 5 6
4n +8
4
3x +4
2
= n 3
+1=
5x +10
8
3p +6
3
10y −2
3
= p 2
+3=
10y +1
9
7u −1
4
−1=
4u +8
5
3v −6
2
+5=
11v −4
5
0.4x + 0.6 = 0.5x − 1.2 0.7x + 0.4 = 0.6x + 2.4
0.9x − 1.25 = 0.75x + 1.75 1.2x − 0.91 = 0.8x + 2.29
0.05n + 0.10(n +8) = 2.15 0.05n + 0.10(n +7) = 3.55
0.10d + 0.25(d +5) = 4.05 0.10d + 0.25(d +7) = 5.25

2L +2(L − 2.5) = 74.
0.49s + 0.21(s − 8) = 22.82
3−7(y −4)=38?
0.35x + 2.1 = 3.85,
a
b

s =

7·4+8 = ? 36
28 + 8 = ? 36
36 = 36 ✓
n =1 st
n −9=2 nd

−12 −3?
−3−9 = ? −12
−12 = −12 ✓
−15?
−3 + (−12) = ? −15
−15 = −15 ✓
−3 −12.
1, 2, 3, 4
−10, −9, −8, −7
150, 151, 152, 153
n +1. n +1, n +1+1, n +2.
n
1 st integer
n +1
2 nd consecutive integer
n +2 3 rd consecutive integer etc.
−54.

n =1 st
n +1=2 nd
n +2=3 rd
−54.
−19 + ⎛ ⎝ −18 ⎞ ⎠ + ⎛ ⎝ −17 ⎞ ⎠ = −54
−54 = −54 ✓
−17, −18, −19.
−96.
−36.
24, 26, 28
−12, −10, −8
n +2.
n +2+2 n +4.
n
1 st even integer
n +2
2 nd consecutive even integer
n +4 3 rd consecutive even integer etc.
63, 65, 67
n, n +2,n +4

n
1 st odd integer
n +2
2 nd consecutive odd integer
n +4 3 rd consecutive odd integer etc.
120
Let n =1 st even integer.
n +2= 2 nd consecutive even integer
n +4= 3 rd consecutive even integer
The sum of the three even integers is 120.
n + n +2+n + 4 = 120
n + n +2+n + 4 = 120
3n + 6 = 120
3n = 114
n =38
1 st integer
n +2
38 + 2
40
n +4
38 + 4
42
2 nd integer
3 rd integer
38 + 40 + 42 = ? 120
120 = 120 ✓
−24.

h +2h − 16,000 = 110,000
h +2h − 16,000 = 110,000
3h − 16,000 = 110,000
3h = 126,000
h = 42,000
2h − 16,000 amount wife earns
2(42,000) − 16,000
84,000 − 16,000
68,000
a b c

a
b
c
a b c
a b a

a =
1 4
1 4
p =
25%
25%
25%

change = new amount − original amount
change is what percent of the original amount?
46 − 36 = 10
p =
27.8%

amount of discount = discount rate · original price
sale price = original amount – discount price
amount of mark-up = mark-up rate · original price
list price = original cost + mark-up
a
b
a
m =
b
p =

a b
a b
I = interest
I = Prt where P = principal
r = rate
t = time
I = Prt.
I = ?
P = $950
r = 3%
t = 5 years

Let I = interest.
I = Prt
I = (950)(0.03)(5)
I = 142.5
6.5%.
I = $1500
P = $7500
r = ?
t = 5 years

Let r = rate of interest.
I = Prt
1,500 = (7,500)r(5)
1,500 = 37,500r
0.04 = r
4% = r
I = Prt
1,500 = ? (7,500)(0.04)(5)
1,500 = 1,500 ✓
I = 4,866.25
P = ?
r = 8.5%
t = 5 years

Let P = principal borrowed.
I = Prt
4,866.25 = P(0.085)(5)
4,866.25 = 0.425P
11,450 = P
I = Prt
4,866.25 = ? (11,450)(0.085)(5)
4,866.25 = 4,866.25 ✓

−36.
−3.
−213.
−267.
a
b
a
a
b
a
a
b
a
a
b
a

a b
a b
a b

a
b

2(x + 3) x = 5.
1 bh b = 14 h = 9.
2
V = 1 3 πr2 h
V = 1 3 πr2 h
πr 2 .

h = 3V
πr 2.
A = 1 bh
2
A = 1 bh
2
C = 5 (F − 32)
9
F = 9 C +32
5
F = 9 C + 32
5
A = 1 h(b + B)
2
S =2πr 2 +2πrh
2πr 2
2πr.

A = P + Prt
A = P + Prt
8x +7y =15
6x
4x + 7y = 9
5x + 8y = 1
A = 1 bh,
2

h =
A = 1 2 bh
126 = 1 2 ·18·h
126 = 9h
14 = h
A = 1 2 bh
126 = ? 1 ·18·14 2
126 = 126 ✓

180°.
m ∠ A + m ∠ B + m ∠ C = 180.
90°
Let a = 1 st angle.
a +40 = 2 nd angle
90 = 3 rd angle (the right angle)
25 + 65 + 90 = ? 180
180 = 180 ✓
25°, 65°, 90°.

a 2 + b 2 = c 2
a 2 +12 2 = 13 2
a 2 + 144 = 169
a 2 = 25
a = 25
a = 5

P =2L +2W
W
w =
2w +6=
P =96

P = 2L +2W
96 = ? 2·34+2·14
96 = 96 ✓
P = a + b + c,

Let x = length of 1 st side.
x + 3 = length of 2 nd side
2x + 2 = length of 3 rd side
29 = ? 6+9+14
29 = 29 ✓

w +40=
P = 2L +2W
360 = ? 2(110) + 2(70)
360 = 360 ✓

x =
a 2 + b 2 = c 2
x 2 + x 2 =10 2
2x 2 = 100
x 2 = 50
x = 50
x ≈ 7.1
a 2 + b 2 = c 2
(7.1) 2 + (7.1) 2 ≈ 10 2 Yes.

C = πd C = πd
π.
V = LWH
V = LWH
A = 1 2 d 1 d 2 d 1 .
A = 1 2 bh
A = 1 2 d 1 d 2 d 2 .
A = 1 2 bh
A = 1 2 h⎛ ⎝b 1 + b 2
⎞
⎠ b 1 .
A = 1 2 h⎛ ⎝b 1 + b 2
⎞
⎠ b 2 .
h =54t + 1 2 at2
h =48t + 1 2 at2
180 = a + b + c 180 = a + b + c
A = 1 pl + B
2
A = 1 pl + B
2
P =2L +2W
P =2L +2W
8x + y =15
9x + y =13
−4x + y =−6
−5x + y =−1
x − y =−4
x − y =−3
4x +3y =7
3x +2y =11
2x +3y =12
5x +2y =10
3x −2y =18
4x −3y =12
20°

C = 5 (F −32)
9
F = 9 C +32
5

a
b
c
a
b

0.25x + 0.10(x + 4).
number · value = total value
23 · $0.10 = $2.30
number · value = total value

p =
8p +3=
$3.02?
7(0.01) + 59(0.05) = ? 3.02
3.02 = 3.02 ✓

3x −5=
10(0.35) + 25(0.49) = ? 15.75
3.50 + 12.25 = ? 15.75
15.75 = 15.75✓

40 − f =
26 · 32 = 832
14 · 26 = —— 364
1,196 ✓

x =
25 − x =
4.5(10) + 8(15) = ? 25(6.60)
45 + 120 = ? 165
165 = 165 ✓

D = rt,

+7
r + 7 = Dennis’ speed
r = Wayne’s speed
Dennis 28 mph(1.5 hours) = 42 miles
Wayne 21 mph(2 hours) = 42 miles ✓

+ 15.

+ 15.
Carina drove 65 mph(3 hours) = 195 miles
Her brother drove 50 mph(4 hours) = ————— 200 miles
395 miles ✓

Now solve this equation 70t +60t = 325
130t = 325
t = 2.5
2.5
Truck going West 70 mph(2.5 hours) = 175 miles
Truck going East 60 mph(2.5hours) = ————— 150 miles
325 miles ✓

+3.
30 minutes is 30
60 or 1 2 hour
15 minutes is 15
60 or 1 4 hour

Walk 3 mph (0.5 hour) = 1.5 miles
Bike 6 mph (0.25 hour) = 1.5 miles

Run 10 mph (1.5 hours) = 15 mi
Bike 16 mph (2.25 hours) = 36 —— mi
51 mi

a
b
c
a
b
c
d
a
b

15 > x.
x >3
x >3.
x >3
x >3 (3, ∞). ∞
x > 3
x ≤1
x =1.
x ≤1 (−∞, 1]. −∞
x ≤1

a x ≥−3b x < 2.5 c x ≤− 3 5
a
−3, −3.
b
c
− 3 5 , −3 5 .
a x > 2 b x ≤ −1.5 c x ≥ 3 4 .
a x ≤ −4 b x ≥ 0.5 c x < − 2 3 .

2.5, 3, 3 2 ,4,4,99?
3
2

ax + b < c, ax + b ≤ c,
ax + b > c, ax + b ≥ c.
a ≠0
ax + b < c, ax + b ≤ c, ax + b > c, ax + b ≥ c.
a < b, then
a + c < b + c
a + c > b + c
a− c < b − c
a− c > b − c

multiply or divide by a positive
if a < b and c > 0, then ac < bc and a c < b c .
if a > b and c > 0, then ac > bc and a c > b c .
multiply or divide by a negative
if a < b and c < 0, then ac > bc and a c > b c .
if a > b and c < 0, then ac < bc and a c < b c .
x > a has the same meaning as a < x
a x − 3 8 ≤ 3 4 b 9y

c
5 3 .
5 3
a p − 3 4 ≥ 1 6 b 9c > 72 c 24 ≤ 3 8 m
a r − 1 3 ≤ 7 12 b 12d ≤ 60 c −24 < 4 3 n
a −13m ≥65b
−2 n ≥8
a
−13.
−13
b

−2.
−2
a −8q < 32 b
k
−12 ≤ 15.
a −7r ≤ − 70 b u −4 ≥ −16.
6y ≤11y + 17,
11y
−5,
3q ≥ 7q − 23.

6x < 10x + 19.
8p +3 ⎛ ⎝p −12 ⎞ ⎠ >7p − 28,
8p +3(p −12)>7p −28
7p
11 > 7.
8p +3p −36>7p −28
11p −36>7p −28
11p −36−7p >7p −28−7p
4p − 36 > −28
4p −36+36>−28+36
4p >8
4p
4 > 8 4
p >2
(2, ∞)
9y + 2 ⎛
⎝y
+ 6 ⎞ ⎠ > 5y − 24
6u + 8(u − 1) > 10u + 32

8x −2(5−x)

1 4 x − 12 1 x > 1 6 x + 7
8
2 5 z − 1 3 z < 15 1 z − 3
5
> ≥ < ≤
Twenty-seven less than x is at least 48.

Let n = the number of tablets.
254.12n ≤ 4000
n ≤ 15.74
n ≤15

Let t = the number of text messages.
28.80 + 0.20t ≤50
0.2t ≤ 21.2
t ≤ 106 text messages
Yes, 28.80 + 0.20(106) = 50.

Let j = the number of invitations.
2.50j − 650 ≥ 2,800
2.5j ≥ 3,450
j ≥ 1,380 invitations

Let h = the number of hours.
525 + 780 + 95(6) ≤ 840 + 45h
1,875 ≤ 840 + 45h
1,035 ≤ 45h
23 ≤ h
h ≥23
1,875 ≤ 840 + 45h
1,875 ≤ 840 + 45(23)
1,875 ≤ 1875

a x 3
b x ≤ −0.5
c x ≥ 1 3
a x ≥−4
b x < 2.5
c x >− 3 2
a x ≤5
b x ≥ −1.5
c x

6h −4(h −1) ≤7h −11 4k − (k −2) ≥7k −26
8m −2(14 − m) ≥7(m −4) +3m
6n −12(3−n) ≤9(n −4) +9n
3
4
b − 1 3 b < 5 12 b − 1 2
9u +5(2u −5) ≥12(u −1) +7u
2
3
g − 1 2 ⎛ ⎝g −14 ⎞ ⎠ ≤ 1 6 ⎛ ⎝g +42 ⎞ ⎠
4
5 h − 2 3 (h −9) ≥ 1 15 (2h +90) 5
6
a − 1 4 a > 7 12 a + 2 3
12v +3(4v −1) ≤19(v −2) +5v
15k ≤ −40 35k ≥ −77
23p −2 ⎛ ⎝6−5p ⎞ ⎠ >3 ⎛ ⎝11p −4 ⎞ ⎠
18q −4 ⎛ ⎝10 − 3q ⎞ ⎠ −61 m
18
≥−4
n
13
≤−6
−22.
−7.

−5x > 10.
n −3 < 12.
a
b

2 (x + 10).
5
−(x − 4).
x + 3 > −4 and 4x −5≤3
2(y +1)−3 x ≤2,

−3 2
−3
6x −3

3x − 4 < 5
4x + 9 ≥ 1.
3(2x +5) ≤18 2(x −7)

1 x −4≥−2 −2(x −3) ≥4.
3
1
3 x −4≥−2 −2(x −3) ≥4
1
3
x −4≥−2 −2x +6≥4
1
3 x ≥2 −2x ≥−2
x ≥6 x ≤1
1
4
x − 3 ≥ −1
−3(x − 2) ≥ 2.
1
5
x − 5 ≥ −3
−4(x − 1) ≥ −2.
a < x x < b
a < x < b
a < x < b. a < x x < b.
Other forms:
a < x < b is equivalent to a < x and x < b
a ≤ x ≤ b is equivalent to a ≤ x and x ≤ b
a > x > b is equivalent to a > x and x > b
a ≥ x ≥ b is equivalent to a ≥ x and x ≥ b

−4≤3x −7

5−3x ≤−1 8+2x ≤5
5−3x ≤−1
−3x ≤−6
8+2x ≤5
2x ≤−3
x ≥2 x ≤− 3 2
⎛
⎝ −∞, − 3 ⎤
2⎦ ∪ [2, ∞)
1 − 2x ≤ −3
7 + 3x ≤ 4.
2 − 5x ≤ −3
5 + 2x ≤ 3.
2 3 x −4≤3 1 (x +8) ≥−1.
4
2
3 x −4≤3 1
4
(x +8) ≥−1
3 ⎛ ⎝ 2 3 x −4⎞ ⎠ ≤3(3) 4·14 (x +8) ≥4·(−1)
2x −12≤9
2x ≤21
x +8≥−4
x ≥ −12
x ≤ 21 2

x ≤ 21 2
x ≥ −12
(−∞, ∞)
3
5
x − 7 ≤ −1
1(x + 6) ≥ −2.
3
3
4
x − 3 ≤ 3
2(x + 10) ≥ 0.
5

x =
24.72 + 1.54x.

x −2 x ≥−4 x ≤−1
x >−6 x

1x −3≤4
3x +2≤−1
2 4
1
3 (x −6) ≥−2 1(x +8) ≥−3
2
3x +7≤1
2x +3≥−5
6(2x −1) >6
5(x +2) ≥0
4−7x ≥−3
5(x −3) +8>3
1x −5≤3
−5≤2x −1

2−7x ≥−5 4(x −3) +7>3.
a
b

−|7|.
, =
a |−8|___ − |−8| b 12___ − |−12| c |−6|___ − 6 d −(−15)___ − |−15|
14 − 2|8 − 3(4 − 1)|.
|n| |n| ≥0
−5 |−5| =5.
5 |5| =5.
−5
|x| =5,
−5

If |x| =5
then x =−5 or x =5
x =±5.
if
then
|u| = a
u =−a or u = a
a |x| =8b |y| =−6c |z| =0
a
b
|x| =8
x =−8or x =8
x =±8
|y| =−6
c
|z| =0
z =−0orz =0
z =0
a |x| = 2 b |y| = −4 c |z| = 0
a |x| = 11 b |y| = −5 c |z| = 0

|5x −4| −3=8.
|3x − 5| − 1 = 6.
|4x − 3| − 5 = 2.
2|x −7| +5=9.

2|x −7| +5=9
2|x −7| =4
|x −7| =2
x −7=−2 x −7=2
x =5 x =9
3|x − 4| − 4 = 8.
2|x − 5| + 3 = 9.
| 2 3 x −4 | +11=3.
| 2 x −4
3 | +11=3
| 2 3 x −4 | =−8
| 3 4 x − 5 |
+ 9 = 4.
| 5 6 x + 3 |
+ 8 = 6.
|u| = |v|
|x −3| = |2x +1|.
|u| = a, u =−a u = a.

if |u| = |v|
then u =−v or u = v
if |u| = |v|
then u =−v or u = v
|5x −1| = |2x +3|.
|5x −1| = |2x +3|
5x −1 = −(2x +3) or 5x −1 = 2x +3
5x −1 = −2x − 3 or 3x −1 = 3
7x −1 = −3 3x = 4
7x = −2 x = 4 3
x = − 2 7
or x = 4 3
|7x − 3| = |3x + 7|.
|6x − 5| = |3x +4|.
|x| =5, −5
|x| =5
x =−5 or x =5
|x| ≤5? −5
−5

|u| ≤ a, −a ≤ u ≤ a.
< ≤
if |u| < a, then −a < u < a
if |u| ≤ a, then −a ≤ u ≤ a
|x|

|5x −6| ≤4
−4 ≤ 5x −6≤4
2≤5x ≤10
2
5 ≤ x ≤2
⎡
⎣
2
5
,2 ⎤ ⎦
|2x − 1| ≤ 5.
|4x − 5| ≤ 3.
|u| < a is equivalent to −a < u < a
|u| ≤ a is equivalent to −a ≤ u ≤ a
|x| ≤5.
−5 −5
|x| ≥5.
−5

−5
|u| ≥ a, u ≤−a u ≤ a.
if |u| > a, then u a
if |u| ≥ a, then u ≤−a or u ≥ a
|x| >4.
|x| >4
x 4
(−∞, −4) ∪ (4, ∞)
−6,
|x| > 2.
|x| > 1.

|2x −3| ≥5.
|2x −3| ≥5
2x −3≤−5 or 2x −3≥5
2x ≤−2 or 2x ≥8
x ≤−1 or x ≥4
(−∞, −1] ∪ [4, ∞)
|4x −3| ≥ 5.
|3x − 4| ≥ 2.
|u| > a is equivalent to u < − a or u > a
|u| ≥ a is equivalent to u ≤ −a or u ≥ a
|actual-ideal| ≤ tolerance
0.075

|actual-ideal| ≤ tolerance
|x −60| ≤ 0.075
−0.075 ≤ x − 60 ≤ 0.075
59.925 ≤ x ≤ 60.075

a |x| =6 b |y| =−3 c
|z| =0
a |x| =4 b |y| =−5 c
|z| =0
a |x| =7 b |y| = −11 c
|z| =0
a |x| =3 b |y| =−1 c
|z| =0
|2x −3| −4=1 |4x −1| −3=0
|3x −4| +5=7 |4x +7| +2=5 4|x −1| +2=10
3|x −4| +2=11 3|4x −5| −4=11 3|x +2| −5=4
−2|x −3| +8=−4
−3|x −4| +4=−5
| 3 4 x −3 | +7=2
| 3 5 x −2 | +5=2 | 1 2 x +5 | +4=1 | 1 4 x +3 | +3=1
|3x −2| = |2x −3| |4x +3| = |2x +1| |6x −5| = |2x +3|
|6−x| = |3−2x|
|x| 5 |x +3| ≥5
|x −7| ≥1 3|x| +4≥1 5|x| +6≥1
2|x +6| +4=8 |6x −5| = |2x +3| |3x −4| ≥2

|2x −5| +2=3 |4x −3|

a ≠0, ax + b =0.

change = new amount − original amount
change is what percent of the original amount?
amount of discount = discount rate · original price
sale price = original amount − discount
amount of mark-up = mark-up rate · original cost
list price = original cost + mark up
I = interest
I = Prt where P = principal
r = rate
t = time
number · value = total value

x > a x ≥ a x < a x ≤ a
a ≠0:
ax + b < c, ax + b ≤ c, ax + b > c, ax + b ≥ c.
a < b, then
a + c < b + c
a + c > b + c
a− c < b − c
a− c > b − c
multiply or divide by a positive
if a < b and c > 0, then ac < bc and a c < b c .
if a > b and c > 0, then ac > bc and a c > b c .
multiply or divide by a negative
if a < b and c < 0, then ac > bc and a c > b c .
if a > b and c < 0, then ac < bc and a c < b c .
> ≥ < ≤

a < x < b a < x x < b.
a < x < b is equivalent to a < x and x < b
a ≤ x ≤ b is equivalent to a ≤ x and x ≤ b
Other forms:
a > x > b is equivalent to a > x and x > b
a ≥ x ≥ b is equivalent to a ≥ x and x ≥ b
|n| |n| ≥0
if |u| = a
then u =−a or u = a
if |u| = |v|
then u =−v or u = v
< ≤
if |u| < a, then −a < u < a
if |u| ≤ a, then −a ≤ u ≤ a
< ≤
|u| < a is equivalent to −a < u < a
|u| ≤ a is equivalent to −a ≤ u ≤ a
> ≥
if |u| > a, then u a
if |u| ≥ a, then u ≤−a or u ≥ a
> ≥

|u| > a is equivalent to u a
|u| ≥ a is equivalent to u ≤−a or u ≥ a
10x −1=5x, x = 1 5
−12n +5=8n, n =− 5 4
6(x +6) =24 −(s +4) =18
23 − 3 ⎛ ⎝y −7 ⎞ ⎠ =8 1
3
(6m +21) = m −7
4 ⎛ ⎝3.5y + 0.25 ⎞ ⎠ = 365 0.25 ⎛ ⎝q −8 ⎞ ⎠ = 0.1 ⎛ ⎝q +7 ⎞ ⎠
8(r −2) =6(r +10) 5+7(2−5x) =2(9x +1) − (13x −57)
(9n +5) − (3n −7) =20−(4n −2) 2 ⎡ ⎣−16 + 5(8k −6) ⎤ ⎦ =8(3−4k) −32
17y −3 ⎛ ⎝4−2y ⎞ ⎠ =11 ⎛ ⎝y −1 ⎞ ⎠ +12y −1 9u +32=15(u −4) −3(2u +21)
−8(7m +4) =−6(8m +9)
2
5
n − 1 10 = 7 10
3
4
a − 1 3 = 1 2 a + 5 6
1
2
(k +3) = 1 3 (k +16)
5y −1
3
+4=
−8y +4
6
0.8x − 0.3 = 0.7x + 0.2 0.10d + 0.05(d −4) = 2.05

−135.
a
b
a
b
a
b

V = LWH
A = 1 2 d 1 d 2 d 2 . h =48t + 1 2 at2
4x −3y =12
67.5
45°

x 2 −2 < x 15p −30 9h −7(h −1)≤4h −23
5n − 15(4 − n) < 10(n −6)+10n
3
8
a − 1 12 a > 5 12 a + 3 4

x ≤5 x >−3 4x −2≤4
5(3x −2) ≤5
7x −1>−8
4(x +2) 3
4
2(x +3) ≥0
3(x +4) ≤6
1
2
x −3≤4
1(x −6) ≥−2
3
|x| =8 |y| = −14 |z| =0
|3x −4| +5=7 4|x −1| +2=10 −2|x −3| +8=−4

| 1 2 x +5 | +4=1 |6x −5| = |2x +3|
|x| ≤8 |2x −5| ≤3 |6x −5| 6 |x| ≥2 |x −5| >−2
|x −7| ≥1 3|x| +4≥1

−5(2x +1) =45
1
4 (12m +28) =6+2(3m +1) 8(3a +5) −7(4a −3) =20−3a
0.1d + 0.25(d +8) = 4.1
14n −3(4n +5) =−9+2(n −8)
3(3u +2) +4 ⎡ ⎣6−8(u −1) ⎤ ⎦ =3(u −2)
3
4
x − 2 3 = 1 2 x + 5 6
|3x −4| =8 |2x −1| = |4x +3|
x +2y =5
x ≥ −3.5 x < 11 −2 ≤ x

4 1
2

5x − 4 x = −1.
3x − 2y x = 4, y = −3.
8 − 3y = 20.

(x, y)
(0, 0). (0, 0)
(0, 0)
(1, 3)
x =1.
y =3. (1, 3).

x =1 y =3
(1, 3).
(0, 4)
(−2, 0)
(a, 0).
(0, b).
a (−5, 4) b (−3, −4) c (2, −3) d (0, −1) e ⎛ ⎝ 3, 5 ⎞
2⎠ .
a x =−5, y =4,
(−5, 4)
b x =−3, y =−4,

(−3, −4)
c x =2, y =−3, (2, −3)
d x =0, (0, −1)
e x =3, y = 5 ,
2
5 2 ⎛ ⎝ 3, 5 ⎞
2⎠
a (−2, 1) b (−3, −1) c (4, −4) d (−4, 4) e ⎛ ⎝ ⎛ −4, 3
2
⎞
⎠
⎞
a (−4, 1) b (−2, 3) c (2, −5) d (−2, 5) e ⎛ ⎝ ⎛ −3, 5
2
⎞
⎠
⎞
Quadrant I Quadrant II Quadrant III Quadrant IV
⎛
⎝ x, y ⎞ ⎠
⎛
⎝ x, y ⎞ ⎠
⎛
⎝ x, y ⎞ ⎠
⎛
⎝ x, y ⎞ ⎠
⎛
⎝ +, + ⎞ ⎠
⎛
⎝ −, + ⎞ ⎠
⎛
⎝ −, − ⎞ ⎠
⎛
⎝ +, − ⎞ ⎠

Ax + By = C.
Ax + By = C,
y =−3x +5 Ax + By = C.
Ax + By = C
y = −3x +5
Add to both sides. y +3x = −3x +5+3x
Simplify. y +3x = 5
Use the Commutative Property to put it in
Ax + By = C form. 3x + y = 5
y =−3x +5 3x + y =5,
Ax + By = C. Ax + By = C,
Ax + By = C.
A ≥0
(x, y).
(x, y) Ax + By = C,

Ax + By = C
y =2x −3
a
b
(0, −3) (3, 3) (2, −3) (−1, −5)
a
b (0, −3), (3, 3), (2, −3), (−1, −5).

(0, 3), (3, −3), (−1, −5) y =2x −3, (2, −3)
y =2x −3
y = 3x − 1.
a
b
(0, −1) (2, 5)

y = 3x − 1.
a
b
(3, −1) (−1, −4)
y =2x +1

y = 2x − 3.
y = −2x + 4.
x
y = 1 2 x +3.
1 2

1 2
y = 1 2 x +3
(x, y)
(0, 3)
(2, 4)
(4, 5)
y = 1 3 x − 1.
y = 1 4 x + 2.
x =−3.

−3, −3.
−3
x =−3
(x, y)
−3 (−3, 1)
−3 (−3, 2)
−3 (−3, 3)
y =4.
y =4
(x, y)
(0, 4)
(2, 4)
(4, 4)

x = a.
(a, 0).
y = b.
(0, b).
a x =2b y =−1.
a
x =2
(x, y)
(2, 1)
(2, 2)
(2, 3)

y =−1
−1.
y =−1
(x, y)
−1 (0, −1)
−1 (3, −1)
−3 −1 (−3, −1)
a x = 5 b y = −4.
a x = −2 b y = 3.
y =4x y =4?
y =4x

y =4
y =4x y =4
y =−3x y =−3

y = −4x y = −4.
y = 3 y = 3x.

(3, 0) (0, 6)
(4, 0) −3 (0, −3)
(5, 0) −5 (0, 5)
(0, 0) (0, 0)
(a, 0) (0, b)
(a, 0)
(0, b)
(a, 0)
(0, b)

a (4, 0). (4, 0).
(0, 2). (0, 2).
b (2, 0). (2, 0).
(0, −6). (0, −6).
c (−5, 0). (−5, 0).
(0, −5). (0, −5).

y =0
x =0
y =0
x =0
2x + y =8.
y =0 x =0

y =0.
y =0.
(4, 0)
x =0.
x =0.
(0, 8)
(4, 0) (0, 8)
2x + y =8
3x + y = 12.
x + 4y = 8.
– x +2y =6

x – 2y = 4.

– x + 3y = 6.
y = 0
x = 0
4x −3y =12
4x −3y =12
(x, y)
(3, 0)
−4 (0, −4)
(6, 4)

5x − 2y = 10.
3x − 4y = 12.

y =5x
(0, 0).
y =5x
(x, y)
(0, 0)
(1, 5)
−1 −5 (−1, −5)
y = 4x.

y = −x.

a (−4, 2) b (−1, −2) c (3, −5) d (−3, 0) e
⎛
⎝ 5 3 ,2⎞ ⎠
a (−2, −3) b (3, −3) c (−4, 1) d (4, −1) e
⎛
⎝ 3 2 ,1⎞ ⎠
a (3, −1) b (−3, 1) c (−2, 0) d (−4, −3) e
⎛
⎝ 1, 14 ⎞
5 ⎠
a (−1, 1) b (−2, −1) c (2, 0) d (1, −4) e
⎛
⎝ 3, 7 ⎞
2⎠
a b
y = x +2;
y = x −4;
(0, 2); (1, 2); (−1, 1); (−3, −1).
(0, −4); (3, −1); (2, 2); (1, −5).

y = 1 2 x −3;
(0, −3); (2, −2); (−2, −4); (4, 1)
y = 1 3 x +2;
(0, 2); (3, 3); (−3, 2); (−6, 0).
y = x +2 y = x −3 y =3x −1
y =−2x +2 y =−x −3 y =−x −2
y =2x y =−2x y = 1 2 x +2
y = 1 3 x −1 y = 4 3 x −5 y = 3 2 x −3
y =− 2 5 x +1 y =−4 5 x −1 y =−3 2 x +2
y =− 5 3 x +4
a x =4b y =3 a x =3b y =1 a x =−2b y =−5
a x =−5b y =−2
y =2x y =2 y =5x y =5 y =− 1 2 x y =−1 2
y =− 1 3 x y =−1 3

x − y =5 x − y =−4 3x + y =6
x −2y =8 4x − y =8 5x − y =5
2x +5y =10 3x −2y =12
−x +4y =8 x +2y =4 x + y =−3
x − y =−4 4x + y =4 3x + y =3
3x − y =−6 2x − y =−8 2x +4y =12
3x −2y =6 2x −5y = −20 3x −4y = −12
y =−2x y =5x y = x

y =−x
y = 3 2 x y =−2 3 x y =−1 2 x +3
y = 1 4 x −2 4x + y =2 5x +2y =10
y =−1 x =3
y = 1 5 x −2.
4x + y =−4?
y = 2 x −2?
3
a
b

(1 – 4)
(8 − 2) .
0
4
, 4 0 .
15
−3 , −15
3 , −15
−3 .
m = rise run .

m = rise run .
m = rise run .
(0, 5) (3, 3)
(0, 5),
(3, 3)
−2.
m = rise run
m = −2
3
m =− 2 3
− 2 3 .

y =4,
m = rise run
m = 0 3
m =0
y = 4 is 0.
x =3,

m = rise run
m = 2 0
x =3
y = b,
x = a,
a x =8b y =−5.
a x =8
b y =−5
x = −4.
y = 7.
(x, y)

(x, y)
(x 1 , y 1 ) read “x sub 1, y sub 1”
(x 2 , y 2 ) read “x sub 2, y sub 2”
(x 1 , y 1 ) (x 2 , y 2 )
(x 3 , y 3 ), (x 4 , y 4 ),
(2, 3) (7, 6),
x
⎛ 1 , y 1 x
⎝ 2, ⎞⎛ 2 , y 2 ⎞
3 ⎠⎝ 7, 6 ⎠
m = rise run
m = 3 5
y 2 , y 1 ,
m = 6−3
7−2
m = y 2 − y 1
7−2
m = y 2 − y 1
x 2 − x 1
m = y 2 − y 1
x 2 − x m = rise
1 run .
(x 1 , y 1 ) (x 2 , y 2 )
m = y 2 − y 1
x 2 − x 1
.
y of the second point minus y of the first point
over
x of the second point minus x of the first point.

(−2, −3) (−7, 4).
(−2, −3) (−7, 4)
⎛ x 1 , y 1 ⎞
⎝−2, −3⎠ ⎛ x 2 , y 2 ⎞
⎝−7, 4 ⎠
m = y 2 − y 1
x 2 − x 1
y y
x x
m = 4 − (−3)
−7 − (−2)
m = 7 −5
m =− 7 5
m = rise run
m = 7 −5
m =− 7 5
(−3, 4) (2, −1).
(−2, 6) (−3, −4).

(1, −1) m = 3 4 .
m = 3 4 , m = −3
−4
(1, −1) −3, −4.
(2, −2) m = 4 3 .
(−2, 3) m = 1 4 .

m = rise run
y = 1 x +3
2
m = rise run
m = 1 2
(0, 3).

y = 1 x +3
2
(0, b) y = mx + b.
a y =− 4 x −2 b x +3y =9
7
a
b
x +3y =9

a y = 2 5 x − 1 b x + 4y = 8
a y = − 4 x + 1 b 3x + 2y = 12
3
y =−x +4
y = mx + b
y =−x +4
m =−1
(0, 4)
m = −1
1
−1
y = −x − 3
y = −x − 1

Equation
Method
#1 x = 2 Vertical line
#2 y = −1 Horizontal line
#3 −x +2y = 6 Intercepts
#4 4x −3y = 12 Intercepts
#5 y =−x + 4 Slope–intercept
Ax + By = C. y =0 x =0
x = a
y = b
y = mx + b,
Ax + By = C,

a y =5b 4x −5y =20c x =−3d y =− 5 9 x +8
a y =5
5
b 4x −5y =20
Ax + By = C.
c x =−3
−3.
d y =− 5 9 x +8
y = mx + b
a 3x + 2y = 12 b y = 4 c y = 1 5 x − 4 d x = −7.
a x = 6 b y = − 3 x + 1 c y = −8 d 4x − 3y = −1.
4
F = 9 C +32
5
a
b
c
d
a
F = 9 5 C +32
F C =0. F = 9 (0) +32
5
F =32

F = 9 5 C +32
F C = 20. F = 9 (20) +32
5
F =36+32
F =68
c
9
5
,
0° 32°
d
(0, 32)
h = 2s + 50
a
b
c
d

T = 1 n + 40
4
a
b
c
d
C = 0.5m +60
a
b
c
d
a
C = 0.5m +60
C m =0. C = 0.5(0) +60
C =60
b
C = 0.5m +60
C m = 250. C = 0.5(250) +60
C = 185
c

d
(0, 60)
m = 0.5
m = 0.5
1
m = 0.5(100)
1(100)
m = 50
100
C = 4p + 25
a
b
c
d
C = 1.8n + 35
a
b
c
d

m = 2 ,
5
m 1 m 2 m 1 = m 2 .

a 3x −2y =6 y = 3 x +1 b y =2x -3 −6x +3y =−9.
2
a
3x −2y =6
y = 3 2 x +1
y.
−2y =−3x +6
−2y
= −3x +6
−2 −2
y = 3 2 x −3
y
y = 3 2 x −3
y = mx + b
m = 3 2
y = 3 2 x +1
y = 3 2 x +1
y = mx + b
y = 3 2
y-intercept is ⎛ ⎝0, −3 ⎞ ⎠
y-intercept is ⎛ ⎝0, 1 ⎞ ⎠
b
y =2x −3 −6x +3y =−9
y
=2x −3
y.
−6x +3y =−9
3y =6x −9
3y
= 6x −9
3 3
y =2x −3
y
=2x −3
y
y =2x −3
y = mx + b
m =2
y =2x −3
y = mx + b
m =2
y-intercept is (0, −3) y-intercept is (0, −3)

a 2x + 5y = 5 y = − 2 5 x − 4 b y = − 1 x − 1 x + 2y = −2.
2
a 4x − 3y = 6 y = 4 3 x − 1 b y = 3 x − 3 3x − 4y = 12.
4
a y =−4 y =3b x =−2 x =−5.
a y =−4 y =3
y =−4 y =3, (0, −4) (0, 3).
b x =−2 x =−5
x =−2 x =−5, (−2, 0) (−5, 0).
a y = 8 y = −6 b x = 1 x = −5.
a y = 1 y = −5 b x = 8 x = −6.
y = 1 x −1 y =−4x +2,
4

m 1 = 1 4 , m 2 =−4,
−1.
m 1 · m 2
1
4 (−4)
−1
m 1 m 2
m 1 =− 1 m 2
.
−1 m 1 · m 2 =−1.
−1,
a y =−5x −4 x −5y =5b 7x +2y =3 2x +7y =5
a

y =−5x −4
y.
x −5y =5
−5y =−x +5
−5y
−5
y
=
−x +5
−5
= 1 5 x −1
y =−5x −4
y = mx + b
m 1 =−5
y = 1 5 x −1
y = mx + b
m 2 = 1 5
−5 ⎛ ⎝ 1 ⎞
5⎠ =−1,
b
y.
7x +2y =3
2y =−7x +3
2y
2
=
−7x +3
2
y =− 7 2 x + 3 2
y
= mx + b
m 1 =− 7 2
2x +7y =5
7y =−2x +5
7y
7
=
−2x +5
7
y =− 2 7 x + 5 7
y
= mx + b
m 1 =− 2 7
a y =−3x + 2 x − 3y = 4 b 5x + 4y = 1 4x + 5y = 3.
a y = 2x − 5 x + 2y = −6 b 2x − 9y = 3 9x − 2y = 1.

y =3 y =−2 x =−5
x =4
(2, 5), (4, 0) (3, 6), (8, 0) (−3, 3), (4, −5)
(−2, 4), (3, −1) (−1, −2), (2, 5) (−2, −1), (6, 5)
(4, −5), (1, −2) (3, −6), (2, −2)

(2, 5); m =− 1 3
(1, 4) m =− 1 2
(−1, −4) m = 4 3
(−3, −5) m = 3 2
m =− 2 5
−2 m = 3 4
(−4, 2) m =4 (1, 5) m =−3
y =−7x +3 y =4x −10 3x + y =5
4x + y =8 6x +4y =12 8x +3y =12
5x −2y =6 7x −3y =9
y =3x −1 y =2x −3 y =−x +3
y =−x −4 y =− 2 5 x −3 y =−3 5 x +2
3x −2y =4 3x −4y =8
x =2 y =5 y =−3x +4
x − y =5 x − y =1 y = 2 3 x −1
3x −2y = −12
2x −5y = −10

P = 31 + 1.75w
a
b
c
d
P = 28 + 2.54w
a
b
c
d
R = 0.575m +42
a
b
c
d
C = 0.32m +15
a
b
c
d
S = 400 + 0.15c
a
b
c
d
S = 750 + 0.09c
a
b
c
d
C = 450 + 28g
a
b
c
d
C = 750 + 42g
a
b
c
d

y = 3 4 x −3; 3x −4y =−2 3x +4y =−2; y = 3 2x −4y =6; x −2y =3
4 x −3
8x +6y =6; 12x +9y =12
x =5; x =−6 x =−3; x =−2
4x −2y =5; 3x +6y =8
8x −2y =7; 3x +12y =9
3x −6y = 12; 6x −3y =3
9x −5y =4; 5x +9y =−1
7x −4y =8; 4x +7y =14
5x −2y = 11; 5x − y =7
3x −2y =8; 2x +3y =6 2x +3y =5; 3x −2y =7 3x −2y =1; 2x −3y =2
2x +4y =3; 6x +3y =2 y =2; y =6 y =−1; y =2
m = 1 2
m =2?
a
b

y-intercept
2 (x + 15).
5
−3 ⎛
⎝x
− (−2) ⎞ ⎠.
y − 3 =−2(x + 1).
y = mx + b.
−9 (0, −4).
y = mx + b.

y = mx + b.
2 5
(0, 4).
−1 (0, −3).
y = mx + b.
(0, −4) (3, −2).
y = mx + b.

(x 1 , y 1 )
(x, y).

m = y − y 1
x − x 1
x − x 1 . m(x − x 1 ) = ⎛ ⎝y − y 1
x − x 1
⎞
⎠ (x − x 1 )
m(x − x 1 ) = y − y 1
y y − y 1 = m(x − x 1 )
(x 1 , y 1 )
y − y 1 = m(x − x 1 )
m =− 1 3
(6, −4).
m = − 2 5
(10, −5).
m = − 3 , (4, −7).
4

y − y 1 = m(x − x 1 ).
(−2, −6).
y − y 1 = m(x − x 1 ).
y − y 1 = m(x − x 1 ).
y =0x −6.
y = a?
(−3, 8).
(−1, 4).
(−3, −1) (2, −2)

(−2, −4) (1, −3).
(−4, −3) (1, −5).
m = y 2 − y 1
x 2 − x 1
y − y 1 = m(x − x 1 ).
(−3, 5) (−3, 4).

(−3, 5) (−3, 4). m = y 2 − y 1
x 2 − x 1
m =
m = −1
0
4−5
−3 − (−3)
−2. x =−2.
(5, 1) (5, −4).
(−4, 4) (−4, 3).
y = mx + b
y − y 1 = m(x − x 1 )
y − y 1 = m(x − x 1 )
y =2x −3. (−2, 1).

y =2x −3.
m ∥ =2. m ∥
(−2, 1) m =2.
(−2, 1)
m =2 m = 2 1
(−2, 1)?
y =2x −3 (−2, 1).

y = 3x + 1 (4, 2).
y = 1 x − 3 (6, 4).
2
y − y 1 = m(x − x 1 ).
y =2x −3. (−2, 1).
m ⊥
⊥
y =2x − 3 perpendicular line
m =2 m ⊥ =− 1 2
(−2, 1) m ⊥ =− 1 2 .
(−2, 1) −1

(−2, 1)?
y =2x −3 (−2, 1).
y = 3x + 1 (4, 2).

y = 1 x − 3 (6, 4).
2
y − y 1 = m(x − x 1 ).
x =5 (3, −2).
x =5.
m ⊥ =0.
(3, −2)
m ⊥ =0
y − y 1 = m(x − x 1 ). y − y 1 = m(x − x 1 )
y − (−2) =0(x −3)
y +2=0
y =−2
x = 4 (4, −5).
x = 2 (2, −1).
x =5 (3, −2). x =5
(3, −2).

(3, −2).
−2.
x =5 y =−2.
y =−3 (−3, 5).
y =−3 x = a.
(−3, 5), −3.
x =−3
y = 1 (−5, 1).

y = −5 (−4, −5).

y (0, 5)
(0, −6)
−3
y (0, −1)
−1
y (0, 3)
1 5
y (0, −5)
− 3 4
y (0, −2)
y (0, −1)
−4
y (0, 0)

m = 5 8 , (8, 3) m = 5 6 , (6, 7) m =−3 , (10, −5)
5
m =− 3 4 , (8, −5) m =−3 2 , (−4, −3) m =−5 , (−8, −2)
2
m =−7, (−1, −3) m =−4, (−2, −3)
(−2, 5)
(−2, −3)
(−1, −7)
(4, −8)

(2, 6) (5, 3) (4, 3) (8, 1) (−3, −4) (5−2).
(−5, −3) (4, −6). (−1, 3) (−6, −7). (−2, 8) (−4, −6).
(0, 4) (2, −3). (0, −2) (−5, −3). (7, 2) (7, −2).
(−2, 1) (−2, −4). (3, −4) (5, −4). (−6, −3) (−1, −3)
y =4x +2,
y =−3x −1,
2x − y =6,
(1, 2)
(2, −3).
(3, 0).
2x +3y =6,
(0, 5).
x =−4,
(−3, −5).
x −2=0,
(1, −2)
y =5,
(2, −2)
y +2=0,
(3, −3)
y =−2x +3,
y =−x +5,
y = 3 4 x −2,
(2, 2)
(3, 3)
(−3, 4)
y = 2 3 x −4,
(2, −4)
2x −3y =8,
(4, −1)
4x −3y =5,
(−3, 2)
2x +5y =6,
(0, 0)
4x +5y =−3,
(0, 0)
x =3,
(3, 4)
x =−5,
(1, −2)
x =7,
(−3, −4)
x =−1,
(−4, 0)
y −3=0,
(−2, −4)
y −6=0,
(−5, −3)
(3, 4)
(2, 1)

(4, 3)
(8, 1)
(−2, 0) (−3, −2)
m = 1 ,
6
(6, 1)
m = 5 ,
6
(6, 7)
4x +3y =6,
(0, −3)
2x +3y =6,
(0, 5)
m =− 3 4 ,
(8, −5)
m =− 3 5 ,
(10, −5)
y −1=0, (−2, 6)
(−6, 2)
x =−3,
(−2, −1)
x =−4,
(−3, −5)
(−3, −4) (2, −5)
(−5, −3) (4, −6)
x −2y =5, (−2, 2)
4x +3y =1, (0, 0)
a
b

x > 2
4x + 3 > 23
8 < x > 3
Ax + By > C Ax+ By ≥ C Ax+ By < C Ax+ By ≤ C
x >3
(x, y)
(x, y)

y > x +4:
a (0, 0) b (1, 6) c (2, 6) d (−5, −15) e (−8, 12)
a
(0, 0)
(0, 0) y > x +4.
b
(1, 6)
(1, 6) y > x +4.
c
(2, 6)
(2, 6) y > x +4.
d
(−5, −15)
(−5, −15) y > x +4.
e

(−8, 12)
(−8, 12) y > x +4.
y > x − 3 :
a (0, 0) b (4, 9) c (−2, 1) d (−5, −3) e (5, 1)
y < x + 1 :
a (0, 0) b (8, 6) c (−2, −1) d (3, 4) e (−1, −4)
x =3
x > 3
x = 3.
y = x +4 y < x +4.
y > x +4. y = x +4
Ax + By = C Ax + By > C
Ax + By < C.
Ax + By < C
Ax+ By ≤ C
Ax + By > C
Ax+ By ≥ C
Boundary line is Ax + By = C
Boundary line is Ax + By = C
Boundary line is not included in solution. Boundary line is included in solution.
Boundary line is dashed.
Boundary line is solid.
y = x +4,

y > x +4
y > x +4?
(1, 6) (−8, 12) y > x +4.
y = x +4.
(0, 0) (−5, −15) y = x +4,
y > x +4. y < x +4.
(2, 6)? 6=2+4, y = x +4,
y > x +4. (2, 6)
y > x +4.
(0, 10)
(0, 10) y > x +4.
y > x +4
10 > ? 0+4
10 > 4
y > x +4.
(0, 0) (−5, −15), y > x +4,

y > x +4
y = x +4 y > x +4.
y = x +4, y > x +4,

y =2x −1.
y =2x −1 y >2x −1
y 2x −1 y 2x −1
y ? 2·0−1 0< ? 2·0−1
0 > −1 True 0 < −1 False
y >2x −1 (0, 0),
y >2x −1.
y ≥2x −1.
(0, 0)?
y

y =−2x + 3.
y = 1 2 x − 4.
2x +3y =6.

2x +3y =6 2x +3y >6
2x +3y 6 2x +3y 6
2(0) +3(0) > ? 6
0 > 6 False
(0, 0) 2x +3y 6.
2x +3y

3x − y = 6.
y ≥ 3 4 x −2.

y ≥ 5 2 x − 4.
y ≤ 2 3 x − 5.
≤ or ≥ ,
< or > ,
x −2y

(0, 0)
(0, 0) x −2y

y ≤−4x.
y =−4x. m =−4 b =0.
≤
(1, 0)
(1, 0) y ≤−4x?
(1, 0) y ≤−4x,
y ≤−4x.
y > − 3x.
y ≥ −2x.

x = a
y = b
vertical line
horizontal line
y >3.
y =3.
(0, 0).
>
(0, 0) y >3.
y >3
0>3
(0, 0)
y >3.
y < 5.
y ≤ −1.
a x

c (x, y)
a
b
10x +15y ≥ 240
15y ≥ −10x + 240
y ≥ − 2 3 x +16
c (15, 10), (0, 16), (24, 0)

a
b
c
a
b
c

y > x −1:
y > x −3:
y 2:
a (−3, 0)
b (1, 6)
c (−6, −2)
d (0, 1)
e (5, −4)
a (5, 1)
b (−2, 6)
c (3, 2)
d (10, −5)
e (0, 0)
a (1, 1)
b (4, −3)
c (0, 0)
d (−8, 12)
e (3, 0)
y =3x −4.
y =2x −4.
y = 1 2 x +1.

y =− 1 3 x −2.
x + y =5.
x + y =3.
3x − y =6.
2x − y =4.
y > 2 3 x −1.
y < 3 5 x +2.
y ≤− 1 2 x +4.
y ≥− 1 3 x −2.
x − y ≤3.
x − y ≥−2.
4x + y >−4.
x +5y 4x.
y ≤−3x.
y < − 10.
y ≥2.
x ≤5.
x ≥0.
x − y

y ≥ 3 2 x.
y ≤ 5 4 x.
y >−2x +1.
y 10.
4x −3y > 12.
$16.50
a
b
c (x, y)
a
b
c (x, y)
a
b
c
a
b
c
>

a
b

3x − 5 x = −2
2x 2 − x − 3 x = a.
7x − 1 − 4x + 5.
(Student name, Student ID #)
(x, y).
{(1, 1), (2, 4), (3, 9), (4, 16), (5, 25)} :
a
b
{(1, 1), (2, 4), (3, 9), (4, 16), (5, 25)}
a {1, 2, 3, 4, 5}

{1, 4, 9, 16, 25}
{(1, 1), (2, 8), (3, 27), (4, 64), (5, 125)} :
a
b
{(1, 3), (2, 6), (3, 9), (4, 12), (5, 15)} :
a
b
a b
c
a
b
c

a b
c
a b
c
a b c
a {(1, 5), (−3, −1), (4, −2), (0, 3), (2, −2), (−3, 4)}.
b
{−3,0,1,2,4}.
−3
c {−2, −1, 3, 4, 5}.
−2

a b c
a b c
(−3, −1) (−3, 4).
a {(−3, 27), (−2, 8), (−1, 1), (0, 0), (1, 1), (2, 8), (3, 27)}

{(9, −3), (4, −2), (1, −1), (0, 0), (1, 1), (4, 2), (9, 3)}
a {(−3, 27), (−2, 8), (−1, 1), (0, 0), (1, 1), (2, 8), (3, 27)}
{−3,−2,−1,0,1,2,3}.
{27, 8, 1, 0}.
b {(9, −3), (4, −2), (1, −1), (0, 0), (1, 1), (4, 2), (9, 3)}
−3.
{0, 1, 2, 4, 9}.
{−3,−2,−1,0,1,2,3}.
a {(−3, −6), (−2, −4), (−1, −2), (0, 0), (1, 2), (2, 4), (3, 6)}
b {(8, −4), (4, −2), (2, −1), (0, 0), (2, 1), (4, 2), (8, 4)}
a {(27, −3), (8, −2), (1, −1), (0, 0), (1, 1), (8, 2), (27, 3)}
b {(7, −3), (−5, −4), (8, −0), (0, 0), (−6, 4), (−2, 2), (−1, 3)}
a b c
a
b

c
{321-549-3327, 427-658-2314, 321-964-7324, 684-358-7961, 684-369-7231, 798-367-8541}
a b c
a b c
a 2x + y =7b y = x 2 +1 c x + y 2 =3
a 2x + y =7
−2
x =3:

x =3, y =1.
b y = x 2 +1
x =2:
x =2, y =5.
c
x =2.
y =1y =−1
x =2, y =1 y =−1.
a 4x + y = −3 b x + y 2 = 1 c y − x 2 = 2
a x + y 2 = 4 b y = x 2 − 7 c y = 5x − 4
f (x).
y = f (x)
f (x)
f is the name of the function
x is the domain value
f (x) is the range value y corresponding to the value x

y = f (x),
x is the independent variable as it can be any value in the domain
y the dependent variable as its value depends on x
y =4x −5. x =2, x =2
x =2.
x =2
y =4x −5
x =2,
f (x) =4x −5.
x =2.
x =2
f (x)
f (x) =2x 2 +3x −1,
a f (3) b f (−2) c f (a)
a
f (3),
b

c
f (a),
f (x) = 3x 2 − 2x + 1,
a f (3) b f (−1) c f (t)
f (x) = 2x 2 + 4x − 3,
a f (2) b f (−3) c f (h)
f (x) g(x)
g(x) =3x −5,
a g(h 2 ) b g(x +2) c g(x) + g(2)
a
g ⎛ ⎝h 2⎞ ⎠,
h 2
b

g(x +2),
x +2
c
g(x) + g(2),
g(2).
b c g(x +2)=3x +1 g(x) + g(2) =3x −4.
g(x +2)≠g(x) + g(2).
g(x) = 4x − 7,
a g(m 2 ) b g(x − 3) c g(x) − g(3)
h(x) = 2x + 1,
a h(k 2 ) b h(x + 1) c h(x) + h(1)
N(t) =75+10t
a
b N(5).
a
t
b N(5).

t =5.
N(5),
N(t) = 100 + 15t
a
b N(7).
N(t) = 110 +25t
a
b N(14).

a b
{(1, 4), (2, 8), (3, 12), (4, 16), (5, 20) ⎫ ⎭ ⎬
{(1, −2), (2, −4), (3, −6), (4, −8), ⎛ ⎝5, −10)}
{(1, 7), (5, 3), (7, 9), (−2, −3), (−2, 8)} {(11, 3), (−2, −7), (4, −8), (4, 17), (−6, 9)}
a b
c
5′4 ″
18.5 – 24.9
5′11′′
18.5 – 24.9

a b
c
a b
c
{(−3, 9), (−2, 4), (−1, 1),
(0, 0), (1, 1), (2, 4), (3, 9)}
{(9, −3), (4, −2), (1, −1),
(0, 0), (1, 1), (4, 2), (9, 3)}
{(−3, 27), (−2, 8), (−1, 1),
(0, 0), (1, 1), (2, 8), (3, 27)}
{(−3, −27), (−2, −8), (−1, −1),
(0, 0), (1, 1), (2, 8), (3, 27)}

a b
c
a 2x + y =−3
b y = x 2
c x + y 2 =−5
a y =3x −5
b y = x 3
c 2x + y 2 =4
a y −3x 3 =2
b x + y 2 =3
c 3x −2y =6
a 2x −4y =8
b −4 = x 2 − y
c y 2 =−x +5
a f(2) b f (−1) c f (a).
f (x) =5x −3 f (x) =3x +4 f (x) =−4x +2
f (x) =−6x −3 f (x) =x 2 − x +3 f (x) =x 2 + x −2
f (x) =2x 2 − x +3
f (x) =3x 2 + x −2
a g(h 2 ) b g(x + 2) c g(x) +g(2).
g(x) =2x +1 g(x) =5x −8 g(x) =−3x −2
g(x) =−8x +2 g(x) =3−x g(x) =7−5x

f (x) =3x 2 −5x; f (2) g(x) =4x 2 −3x; g(3) F(x) =2x 2 −3x +1;
F(−1)
G(x) =3x 2 −5x +2;
G(−2)
h(t) =2|t −5| +4; h(−4)
h(y) =3|y −1| −3; h(−4)
f (x) =
x x +2 ; f (2) g(x) =x −2
−1 x +2 ; g(4)
N(t) =85+20t
a
b N(4).
N(t) =43+t
a
b N(30).
C(x) = 3.25x + 1500
a
b C(0).
c C(1000).
C(x) = 7.25x + 2500
C(x)
a
b C(0).
c C(1000).
a
b

a 2 3 b 3 2 .
a |7| b |−3|.
a 4 b 16.
(x, y)
y =2x −3,

y =2x −3
a
b

y =2x −3
y =2x −3
f (x) =2x −3.
(x, y) y = f (x). ⎛ ⎝x, f (x) ⎞ ⎠.

y =2x −3
f (x) =2x −3
(x, y)
y = f (x).
f name of function
x x-coordinate of the ordered pair
f (x) y-coordinate of the ordered pair
⎛
⎝x, f (x) ⎞ ⎠
y =2x −3
y = mx + b.
f (x) = mx + b

f (x) =−2x −4.
f (x) =−2x −4
m =−2
b =−4
f (x) = −3x − 1
f (x) = −4x − 5
f (x) = b,

f (x) y = b.
f (x) = b,
f (x) =4.
f (x) =4
(0, 4).
f (x) = −2.
f (x) = 3.
f (x) = x
f (x) =1x +0,

f (x) = x 2 .
f (x) = x 2 .
f (x) = −x 2

f (x) = x 3 .
f (x) = x 3 .
f (x) = −x 3 .

f (x) = x
f (x) =
x
f (x) = x.
f (x) = − x.

f (x) = |x|.
f (x) = |x|.
f (x) = |x|.
f (x) = −|x|.

[−3, 3].
[−1, 3].

a f (0).
b f ⎛ ⎝ 3 2 π⎞ ⎠ .
c f ⎛ ⎝ −1 2 π⎞ ⎠ .
d f (x) =0.
e
f
g
h
a x =0, f (0) =0.

x = 3 2 π, −1. f ⎛ ⎝ 3 2 π⎞ ⎠ =−1.
c x =− 1 2 π, −1. f ⎛ ⎝ −1 2 π⎞ ⎠ =−1.
d (−2π, 0), (−π, 0), (0, 0), (π, 0), (2π, 0). f (x) =0
−2π, −π, 0,π, 2π.
e y =0. f (x) =0.
(−2π, 0), (−π, 0), (0, 0), (π, 0), (2π, 0).
f x =0. f (0). (0, 0).
g (−∞, ∞)
h −1 [−1, 1].
a f (0).
b f ⎛1 ⎞ ⎝ π 2 ⎠.
c f ⎛ ⎝ ⎛ − 3
2
π ⎞ ⎠ ⎞ .
d f (x) = 0.
e
f
g
h

a f (0).
b f (π).
c f (−π).
d f (x) = 0.
e
f
g
h

a
a
b
b
a
a
b
b

a b
f (x) =3x +4 f (x) =2x +5 f (x) =−x −2
f (x) =−4x −3 f (x) =−2x +2 f (x) =−3x +3
f (x) = 1 2 x +1 f (x) = 2 3 x −2 f (x) =5
f (x) =2 f (x) =−3 f (x) =−1
f (x) =2x f (x) =3x f (x) =−2x
f (x) =−3x f (x) =3x 2 f (x) =2x 2
f (x) =−3x 2 f (x) =−2x 2 f (x) = 1 2 x2
f (x) = 1 3 x2 f (x) = x 2 −1 f (x) = x 2 +1
f (x) =−2x 3 f (x) =2x 3 f (x) = x 3 +2
f (x) = x 3 −2 f (x) =2 x f (x) =−2 x
f (x) = x −1 f (x) = x +1 f (x) =3|x|
f (x) =−2|x| f (x) = |x| +1 f (x) = |x| −1

a f (0).
b f ⎛ ⎝ 1 2 π⎞ ⎠ .
c f ⎛ ⎝ −3 2 π⎞ ⎠ .
d f (x) =0.
e
f
g
h
a f (0).
b f (π).
c f (−π).
d f (x) =0.
e
f
g
h

a f (0).
b f (−3).
c f (3).
d f (x) =0.
e
f
g
h
a f (0).
b f (x) =0.
c
d
e
f
a
b

Ax + By = C
Ax + By > C Ax + By < C.
y = b.
(0, b).
Ax + By = C,
Ax + By > C, Ax + By ≥ C, Ax + By < C, Ax + By ≤ C,
(x, y)
(0, 0)
(x 1 , y 1 )
y − y 1 = m(x − x 1 ).
(x, y).
(x, y)
Ax + By = C,
(x, y)
Ax + By = C.
x = a. (a, 0).
(a, 0).
0 (0, b).
Quadrant I Quadrant II Quadrant III Quadrant IV
⎛
⎝ x, y ⎞ ⎠
⎛
⎝ x, y ⎞ ⎠
⎛
⎝ x, y ⎞ ⎠
⎛
⎝ x, y ⎞ ⎠
⎛
⎝ +, + ⎞ ⎠
⎛
⎝ −, + ⎞ ⎠
⎛
⎝ −, − ⎞ ⎠
⎛
⎝ +, − ⎞ ⎠

Ax + By = C
(a, 0)
(0, b)
y =0
x =0
y =0
x =0
m = rise run .

m = rise run .
m = rise run .
(x 1 , y 1 ) (x 2 , y 2 )
m = y 2 − y 1
x 2 − x 1
.
m = rise run
(0, b) y = mx + b
m 1 m 2 m 1 = m 2 .
m 1 m 2
m 1 =−m 1 . 2
−1, m 1 · m 2 =−1.
y − y 1 = m(x − x 1 ).

m = y 2 − y 1
x 2 − x 1
y − y 1 = m(x − x 1 ).
y = mx + b
y − y 1 = m(x − x 1 )
y − y 1 = m(x − x 1 )
y − y 1 = m(x − x 1 ).
y − y 1 = m(x − x 1 )
≤or≥,
,
y = f (x)
f (x)

f (x)
y = f (x),
(x, y)
y = f (x).
f
x
f (x)
name of function
x-coordinate of the ordered pair
y-coordinate of the ordered pair
⎛
⎝x, f (x) ⎞ ⎠

a (−1, −5)
b (−3, 4)
c (2, −3)
d ⎛ ⎝ 1, 5 ⎞
2⎠
a (−2, 0)
b (0, −4)
c (0, 5)
d (3, 0)
5x + y = 10;
y =6x −2;
a (5, 1)
b (2, 0)
c (4, −10)
a (1, 4)
b ⎛ ⎝ 1 3 ,0⎞ ⎠
c (6, −2)
y =4x −3 y =−3x y = 1 2 x +3
y =− 4 5 x −1 x − y =6 2x + y =7
3x −2y =6
y =−2 x =3
y =−2x y =−2 y = 4 3 x y = 4 3

x − y =−1 x +2y =6 2x +3y =12
y = 3 4 x −12 y =3x
−x +3y =3 x − y =4 2x − y =5
2x −4y =8 y =4x

y =2 x =5 x =−3
y =−1
(−1, −1), (0, 5) (3,5), (4, −1) (−5, −2), (3, 2)
(2, 1), (4, 6)

(2, −2); m = 5 (−3, 4); m =−
2
1 3
−4; m =3
m =− 3 4
y =−4x +9 y = 5 3 x −6 5x + y =10
4x −5y =8
y =2x +3 y =−x −1 y =− 2 5 x +3
4x −3y =12
x =5 y =−3 2x + y =5
x − y =2 y = 2 2 x +2 y = 3 4 x −1
C = 6.5m +42
a
b
c
d
P =35h − 250
a
b
c
d
4x −3y =−1;y = 4 3 x −3 y =5x −1;10x +2y =0 3x −2y =5;2x +3y =6
2x − y =8;x −2y =4

1 y
−5 y y
3
(0, −3)
(0, 4)
(0, −6)
−2 y
(0, 0)
m =− 1 4 , (−8, 3) m = 3 , (10, 6)
5
(−2, 7) m =−2, (−1, −3)

(2, 10) (−2, −2) (7, 1) (5, 0)
(3, 8) (3, −4) (5, 2) (−1, 2)
y =−3x +6, (1, −5) 2x +5y = −10, (10, 4)
x =4, (−2, −1) y =−5, (−4, 3)
y =− 4 5 x +2, (8, 9) 2x −3y =9, (−4, 0)
y =3, (−1, −3) x =−5 (2, 1)
y < x −3:
x + y >4:
a (0, 1) b (−2, −4) c (5, 2) d (3, −1)
e (−1, −5)
a (6, 1) b (−3, 6) c (3, 2) d (−5, 10) e (0, 0)
y =−x +2.
y = 2 3 x −3.

x + y =−4.
x −2y =6.
y > 2 5 x −4. y ≤− 1 4 x +3. x − y ≤5.
3x +2y > 10.
y ≤−3x.
y

a b
c
a b
c
a b
c
{(9, −5), (4, −3), (1, −1),
(0, 0), (1, 1), (4, 3), (9, 5)}
{(−3, 27), (−2, 8), (−1, 1),
(0, 0), (1, 1), (2, 8), (3, 27)}
a b
c

2x + y =−3 y = x 2 y =3x −5
y = x 3 2x + y 2 =4
a f (−2) b f(3) c f (a).
f (x) =3x −4 f (x) =−2x +5 f (x) =x 2 −5x +6
f (x) =3x 2 −2x +1
g(x) =3x 2 −5x; g(2) F(x) =2x 2 −3x +1;
F(−1)
h(t) =4|t −1| +2; h(−3)
f (x) = x +2
x −1 ; f (3)

a b
f (x) =5x +1 f (x) =−4x −2 f (x) = 2 3 x −1
f (x) =−6 f (x) =2x f (x) =3x 2
f (x) =− 1 2 x2 f (x) = x 2 +2 f (x) = x 3 −2
f (x) = x +2 f (x) =−|x| f (x) = |x| +1

a f (0).
b f ⎛ ⎝ 1 2 π⎞ ⎠ .
c f ⎛ ⎝ −3 2 π⎞ ⎠ .
d
f (x) =0.
e x
f y
g
h
a f (0).
b
f (x) =0.
c x
d y
e
f

a (2, 5)
b (−1, −3)
c (0, 2)
d ⎛ ⎝ −4, 3 ⎞
2⎠
e (5, 0)
3x − y =6?
a (3, 3) b (2, 0) c (4, −6)
a
(5, 2)
(−1, −4).
1 2
(−3, −4).
b
4x +2y =−8

y = 5 3 x −1 y =−x y =2
− 3 y
m =2, (−3, −1) (10, 1)
4
(6, −1)
(0, −2)
y = 5 4 x +2,
(−10, 3)
y =−x −3.
y > 3 2 x +5 x − y ≥−4
y ≤−5x
a
b
c (x, y)
a
b
c
{(−3, 27), (−2, 8), (−1, 1), (0, 0),
(1, 1), (2, 8), (3, 27)}
f (c).
a f (−1) b f (2) c
f (x) =4x 2 −2x −3

h(y) =3|y −1| −3, h(−4).
a b
f (x) = x 2 +1 f (x) = x +1
b y
c f (−1).
d f (1).
e
f

y = 2 3 x − 4,
a (6, 0) b (−3, −2)

3x − y = 12.
2x − 3y = 12.
⎧2x + y =7
⎨
⎩x −2y =6
2x + y =7,
(x, y)
(x, y).
⎧ x − y =−1
⎨
⎩2x − y =−5 .
a (−2, −1) b (−4, −3)
a

⎧3x + y = 0
⎨
⎩ x + 2y = −5 .
a (1, −3) b (0, 0)
⎧
⎨
⎩
x − 3y = −8
−3x − y = 4 .
a (2, −2) b (−2, 2)
⎧2x + y =7
⎨
⎩x −2y =6 .

⎧x
− 3y = −3
⎨ .
⎩ x + y = 5
⎧
⎨
⎩
−x + y = 1
3x + 2y =12 .
⎧
⎨
⎩
3x + y =−1
.
2x + y =0
y

(−1, 2).
(−1, 2).
⎧
⎨
⎩
−x + y = 1
2x + y = 10 .
⎧2x + y = 6
⎨
⎩ x + y = 1 .
⎧
⎨ y = 1 2 x −3 .
⎩x −2y =4

⎧ y = − 1x + 2
⎨ 4 .
⎩ x + 4y = − 8
⎧ y = 3x − 1
⎨
⎩6x − 2y = 6 .
⎧
⎨
⎩
y =2x −3
−6x +3y =−9 .

⎧ y = − 3x − 6
⎨
⎩6x + 2y = − 12 .
⎧ y = 1x − 4
⎨ 2 .
⎩2x − 4y = 16

a ⎧ =3x −1
⎨y
⎩6x −2y =12 b ⎧ + y =−3
⎨2x
⎩x −5y =5
a
⎧
⎨
⎩
y = 3x −1
6x −2y = 12
y = 3x −1
6x −2y = 12
−2y = −6x +12
−2y
= −6x +12
−2 −2
y = 3x −6
y = 3x −1 y = 3x −6
m = 3 m = 3
b = −1 b = −6
b

⎧2x + y = −3
⎨
⎩x −5y = 5
2x + y = −3 x −5y = 5
y = −2x −3 −5y = −x +5
−5y
−5
=
−x +5
−5
y = 1 5 x −1
y = −2x −3 y = 1 5 x −1
m = −2 m = 1 5
b = −3 b = −1
a ⎨⎧
⎧
⎩
y = −2x −4
4x + 2y = 9 b ⎧ 3x + 2y = 2
⎨⎧
⎩2x + y = 1
a ⎨⎧
⎧ y = 1x − 5 3 ⎩x
− 3y = 6
b ⎧ ⎨⎧
⎩
⎧ x + 4y =12
−x + y = 3
−10
⎧2x + y =7
⎨
⎩x −2y =6

⎧2x + y =7
⎨
⎩x −2y =6 .
⎧−2x + y = −11
⎨
.
⎩ x + 3y = 9
⎧
⎨
⎩
2x + y = −1
4x + 3y = 3 .

⎧
⎨
⎩
4x +2y =4
6x − y =8 .
−2x +2
−2x +2.
x = 5 4 4x +2y =4 ⎛ ⎝ 5 4 , − 1 ⎞
2⎠ .

⎛ ⎝ 5 4 , − 1 ⎞
2⎠ .
⎧
⎨
⎩
x − 4y = −4
−3x + 4y = 0 .
⎧
⎨
⎩
4x − y = 0
2x − 3y = 5 .
if a = b
and c = d
then a + c = b + d.
⎧ 3x + y =5
⎨
⎩—————
2x − y =0
5x =5
⎧ x +4y =2
⎨
⎩2x +5y =−2
−2,
−2.

⎧2x + y =7
⎨
⎩x −2y =6 .
⎧
⎨
⎩
3x + y = 5
2x − 3y = 7 .
⎧
⎨
⎩
4x + y = − 5
−2x − 2y = − 2 .

⎧
⎨
⎩
4x −3y =9
7x +2y =−6 .
x =0
(0, −3).

(0, −3).
⎧
⎨
⎩
3x − 4y = − 9
5x + 3y = 14 .
⎧
⎨
⎩
7x + 8y = 4
3x − 5y = 27 .
⎨ x + 1 2 y =6
3
⎩2 x + 2 3 y = 17 .
2
⎧
y −4.
x =3
y
(3, 6).

(3, 6).
⎧1
x −
3
1 2
⎨
y = 1
3
⎩ x − y =
4
5 .
2
⎧ x + 3y = − 1
⎨ 5 5
⎩− 1 2 x − 2 3 y = 5 .
6
⎧3x +4y =12
⎨
⎩y =3− 3 4 x .

⎧3x +4y =12
⎨
⎩y =3− 3 4 x
⎧3x +4y =12
⎨3
⎩4 x + y =3
⎧3x +4y =12
⎨
⎩4 ⎛ ⎝ 3 4 x + y⎞ ⎠ =4(3)
⎧
⎨
⎩
⎧
⎨
⎩
3x +4y =12
3x +4y =12
3x +4y =12
______________
−3x −4y =−12
0=0
⎧5x − 3y = 15
⎨
⎩ y = − 5 + 5 3 x.
⎧ x + 2y = 6
⎨
⎩ y = − 1 2 x + 3.
Choose the Most Convenient Method to Solve a System of Linear Equations
Graphing
————
Substitution
————— Elimination
—————
Use when you need a Use when one equation is Use when the equations are
picture of the situation. already solved or can be in standard form.
easily solved for one
variable.
a ⎧ +8y =40
⎨3x
⎩7x −4y = −32 b ⎧ +6y =12
⎨5x
⎩y = 2 3 x −1

a
⎧
⎨
⎩
3x +8y =40
7x −4y = −32
b
⎧5x +6y =12
⎨
⎩y = 2 3 x −1
a ⎧ 4x − 5y = −32
⎨⎧
⎩3x + 2y = −1
b ⎨⎧
⎧ x = 2y − 1
⎩3x − 5y = −7
a ⎨⎧
⎧ y =2x − 1
⎩3x − 4y = − 6 b ⎧ 6x − 2y = 12
⎨⎧
⎩3x + 7y = −13

⎧2x −6y =0
⎧−3x + y =8
⎧x + y =2
⎨ ⎨
⎩3x −4y =5
⎩−x +2y =−9
⎨
⎩y = 3 4 x
a (3, 1) b (−3, 4)
a (−5, −7) b (−5, 7)
a ⎛8 ⎝7 , 6 ⎞
7⎠ b ⎛ ⎝ 1, 3 ⎞
4⎠
⎧2x +3y =6
⎨
⎩y = 2 3 x +2
a (−6, 2) b (−3, 4)
⎧3x + y =−3
⎧−x + y =2
⎨ ⎨
⎩2x +3y =5
⎩2x + y =−4
⎧ y = x +2
⎨
⎩y =−2x +2
⎧ y = x −2
⎨
⎩y =−3x +2
⎧
⎨ y = 3 2 x +1
y =− 1 2 x +5
⎩
⎧
⎨ y = 2 3 x −2
y =− 1 3 x −5
⎩
⎧ x + y =−4
⎨
⎩−x +2y =−2
⎧
⎨
⎩
−x +3y =3
x +3y =3
⎧
⎨
⎩
−2x +3y =3
x +3y =12
⎧
⎨
⎩
2x − y =4
2x +3y =12
⎧x +3y =−6
⎨
⎩y =− 4 3 x +4
⎧−x +2y =−6
⎨
⎩y =− 1 2 x −1
⎧−2x +4y =4
⎨
⎩y = 1 2 x
⎧3x +5y =10
⎨
⎩y =− 3 5 x +1
⎧
⎨
⎩
4x −3y =8
8x −6y =14
⎧
⎨
⎩
x +3y =4
−2x −6y =3
⎧
⎨
⎩
x =−3y +4
2x +6y =8
⎧
⎨
⎩
4x =3y +7
8x −6y =14
⎧
⎨
⎩
2x + y =6
−8x −4y = −24
⎧
⎨
⎩
5x +2y =7
−10x −4y = −14
⎧
⎨ y = 2 3 x +1
⎧
⎨ y = 3 2 x +1
⎧5x +3y =4
⎨
⎩2x −3y =5
⎩−2x +3y =5
⎩2x −3y =7

⎧
⎨ y =−1 2 x +5
⎩x +2y =10
⎧5x −2y =10
⎨
⎩y = 5 2 x −5
⎧2x + y =−4
⎧2x + y =−2
⎨ ⎨
⎩3x −2y =−6
⎩3x − y =7
⎧
⎨
⎩
x −2y =−5
2x −3y =−4
⎧
⎨
⎩
x −3y =−9
2x +5y =4
⎧
⎨
⎩
5x −2y =−6
y =3x +3
⎧
⎨
⎩
−2x +2y =6
y =−3x +1
⎧2x +5y =1
⎨
⎩y = 1 3 x −2
⎧3x +4y =1
⎨
⎩y =− 2 5 x +2
⎧
⎨
⎩
2x + y =5
x −2y = −15
⎧
⎨
⎩
4x + y =10
x −2y = −20
⎧y =−2x −1
⎨
⎩y =− 1 3 x +4
⎧
⎨ y = x −6
⎩y =− 3 2 x +4
⎧ x =2y ⎨
⎩4x −8y =0
⎧
⎨
⎩
2x −16y =8
−x −8y =−4
⎧
⎨ y = 7 8 x +4
⎩−7x +8y =6
⎧
⎨ y =−2 3 x +5
⎩2x +3y =11
⎧5x +2y =2
⎧6x −5y =−1
⎨ ⎨
⎩−3x − y =0
⎩2x + y =13
⎧
⎨
⎩
2x −5y =7
3x − y =17
⎧
⎨
⎩
5x −3y =−1
2x − y =2
⎧
⎨
⎩
3x −5y =−9
5x +2y =16
⎧
⎨
⎩
4x −3y =3
2x +5y = −31
⎧
⎨
⎩
3x +8y =−3
2x +5y =−3
⎧
⎨
⎩
11x +9y =−5
7x +5y =−1
⎧
⎨
⎩
3x +8y =67
5x +3y =60
⎧
⎨
⎩
2x +9y =−4
3x +13y =−7
⎧
⎨
⎩
1
3 x − y =−3
x + 5 2 y =2
⎧
⎨ x + 1 2 y = 3 2
1
⎩5 x − 1 5 y =3
⎧
⎨ x + 1 3 y =−1
1
⎩3 x + 1
2 y =1
⎧
⎨
⎩
1
3 x − y =−3
2
3 x + 5 2 y =3
⎧
⎨
⎩
2x + y =3
6x +3y =9
⎧
⎨
⎩
x −4y =−1
−3x +12y =3
⎧
⎨
⎩
−3x − y =8
6x +2y = −16
⎧
⎨
⎩
4x +3y =2
20x +15y =10

a ⎧ −15y = −32
⎨8x a ⎧ =7x −5
⎨y
⎩6x +3y =−5
⎩3x −2y =16
b ⎧ =4y −3
⎨x
⎩4x −2y =−6
b ⎧ −5y = −42
⎨12x
⎩3x +7y = −15
a ⎧ =4x +9
⎨y
⎩5x −2y = −21
b ⎧ −4y =24
⎨9x
⎩3x +5y = −14
a ⎧ −15y = −30
⎨14x
⎩7x +2y =10
b ⎧ =9y −11
⎨x
⎩2x −7y = −27
⎧5x +4y =10
⎨
⎩2x =3y +27 .
⎧3x + y =12
⎨ .
⎩x = y −8
⎧ x + y =10
⎨
⎩x − y =6
a b
c

n =
m =
m = 9 2
9 2 −9 2 .
−6.

$10,000 + $40
s =
n =

e =
c =

Let x = the measure of the first angle.
y = the measure of the second angle
x + y =90
x − y =26
⎧ x + y =90
⎨
⎩x − y =26
⎧ x + y =90
⎨
x − y =26
___________
⎩
2x = 116
x =58
x =58
x + y =90
58 + y =90
y =32
58 + 32 = 90✓
58 − 32 = 26✓

x =
y =

a =
b =
3b +10
b =20

L =
W =

D = rt
D = r · t
65j =78k
1 2

k = j − 1 2
⎧
⎨ k = j − 1 2
⎩65j =78k
k = j − 1 2
j
65j = 78k
65j = 78 ⎛ ⎝ j − 1 ⎞
2⎠
65j = 78j −39
−13j = −39
j = 3
j =3
k
k = j − 1 2
k =3− 1 2
k = 5 2 or k =21 2
3 hours ⎛ ⎝65 mph ⎞ ⎠ = 195 miles
2 1 2 hours ⎛ ⎝ 78 mph⎞ ⎠ = 195 miles
2 1 2
( 1 hour)
4

+ c.
b − c.

s =
c =
13.5 + 1.5 = 15
15 · 4 = 60
13.5 − 1.5 = 12
12 · 5 = 60

j =
w =

347 + 18 = 365
365 · 3 = 1,095
347 − 18 = 329
329 · 3 = 987

−26.
−27. −59.
−45. −89.

4.5
a
b

4.025(1,562).

a ·12=12a,
c ·7=7c.
a =
c =
−7.
a = 521

n =
d =
n(0.05) = 0.05n
d(0.10) = 0.10d.
d =2n −9

n =36
$0.10 = $6.30
$0.05 = $1.80
= $8.10 ✓
n =
c =

Number • Value = Total Value
−2
n =16
16 + 4 = 20 ✓
9 · 16 + 2 · 4 = 152 ✓

x =
y =
−0.5
x =80
80 + 120 = 200 ✓
0.25(80) + 0.50(120) = 200 ✓
Yes!

I = Prt.
7.1%
s =
b =

s + b = 40,000,
0.08s + 0.03b = 0.071(40,000),
6.2%
12.5%
10.5%
5.9%. $1,669.68.
b =
f =

C(x) = ⎛ ⎝cost per unit ⎞ ⎠ · x + fixed costs
R(x) = ⎛ ⎝selling price per unit ⎞ ⎠ · x
C(x) = R(x)

a
b
c
d
a
C(x) = ⎛ ⎝cost per unit ⎞ ⎠ · x + fixed costs
C(x) = 105x + 7000
b
R(x) = ⎛ ⎝selling price per unit ⎞ ⎠ · x
R(x) = 245x
c
⎧C(x) = 105x + 7000 ⎧y = 105x + 7000
⎨ or ⎨
⎩R(x) = 245x
⎩y = 245x
d

C(x) = R(x)
105x + 7000 = 245x
7000 = 140x
50 = x
(50, 12,250).
a
b
c
d
a
b
c
d

$3.80.
$6.20.
$1.80
$1.20
$1.40
$0.95.
$13.10.
$1.79
$2.49
$2.21

10.6%
7.25%
5.5%
4.5%
$2878.50.
5.25%

a
b
c
a
b
c
d
d
⎧ s + b = 40,000
⎨
⎩0.08s + 0.03b = 0.071(40,000) .
a
b

5x − 2y + 3z x = −2, y = −4, z = 3.
⎧−2x + y = −11
⎨
.
⎩ x + 3y = 9
⎧7x + 8y = 4
⎨
⎩3x − 5y = 27 .
ax + by = c,
(x, y),

ax + by + cz = d,
(x, y, z)
ax + by + cz = d
(x, y, z)

(x, y, z)
(x, y, z).
⎧
⎨
⎩
x − y + z =2
2x − y − z =−6 .
2x +2y + z =−3
a (−2, −1, 3) b (−4, −3, 4)
a

a (1, −3, 2) b (4, −1, −5)
⎧
⎨
⎩
3x + y + z = 2
x + 2y + z = −3.
3x + y + 2z = 4
a (2, −2, 3) b (−2, 2, 3)
⎧x
− 3y + z = −5
⎨−3x − y − z = 1 .
⎩2x − 2y +3z = 1
⎧
⎨
⎩
x −2y + z =3
2x + y + z =4 .
3x +4y +3z =−1

⎧
⎨
⎩
3x + y − z = 2
2x − 3y − 2z = 1.
⎩4
x − y − 3z = 0
⎧4x + y + z = −1
⎨−2x − 2y + z = 2.
⎩2x + 3y − z = 1
⎧3x −4z =0
⎨3y +2z =−3.
⎩2x +3y =−5
⎧
⎨
⎩
3x −4z =0 (1)
3y +2z =−3(2)
2x +3y =−5(3)
z
x y x y
x =−4

x =−4 y =1. x =−4
(−4, 1, −3)
3x −4z = 0(1)
3(−4) −4(−3) = ? 0
0 = 0 ✓
3y +2z = −3(2)
3(1) +2(−3) = ? −3
−3 = −3 ✓
2x +3y = −5(3)
2(−4) +3(1) = ? −5
−5 = −5 ✓
The solution is (−4, 1, −3).
⎧3x − 4z = −1
⎨2y + 3z = 2 .
⎩2x + 3y = 6
⎧4x − 3z = −5
⎨3y + 2z = 7 .
⎩3x + 4y = 6
⎧
⎨
⎩
x +2y −3z =−1
x −3y + z =1 .
2x − y −2z =2
⎧
⎨
⎩
x +2y −3z =−1(1)
x −3y + z =1 (2)
2x − y −2z =2 (3)
z

⎧ x + 2y + 6z = 5
⎨−x + y − 2z = 3.
⎩x
− 4y − 2z = 1
⎧
⎨
⎩
2x − 2y + 3z = 6
4x − 3y + 2z = 0 .
−2x + 3y − 7z = 1
⎧
⎨
⎩
x +2y − z =1
2x +7y +4z =11.
x +3y + z =4
⎧
⎨
⎩
x +2y − z =1 (1)
2x +7y +4z =11(2)
x +3y + z =4 (3)
y

y +2z = 3
y = −2z +3
x +2y − z =1
y =−2z +3. x +2(−2z +3)−z = 1
x −4z +6−z = 1
x −5z +6 = 1
x = 5z −5
0=0
(x, y, z) x =5z −5;y =−2z +3
⎧ x + y − z = 0
⎨2x + 4y − 2z = 6.
⎩3x + 6y − 3z = 9
⎧x
− y − z = 1
⎨−x + 2y − 3z = −4.
⎩3x − 2y − 7z = 0
y = 2x
2x − y = 0

y.
x =75
−2x + y =0
x = 75. −2(75) + y = 0
−150 + y = 0
y = 150 student tickets
x + y + z = 250
x = 75, y = 150. 75 + 150 + z = 250
225 + z = 250
z = 25 child tickets

⎧
⎨
⎩
2x −6y + z =3
3x −4y −3z =2
2x +3y −2z =3
⎧
⎨
⎩
−3x + y + z =−4
−x +2y −2z =1
2x − y − z =−1
a (3,1,3)b (4, 3, 7)
a (−5, −7, 4) b (5, 7, 4)
⎧y −10z =−8
⎨2x − y =2
⎩x −5z =3
a (7, 12, 2) b (2, 2, 1)
⎧x +3y − z =15
⎪
⎨y = 2
⎪ 3 x −2
⎩x −3y + z =−2
a ⎛ ⎝ −6, 5, 1 ⎞
2⎠ b ⎛ ⎝ 5, 4 3 ,−3⎞ ⎠
⎧
⎨
⎩
5x +2y + z =5
−3x − y +2z =6
2x +3y −3z =5
⎧
⎨
⎩
6x −5y +2z =3
2x + y −4z =5
3x −3y + z =−1
⎧
⎨
⎩
2x −5y +3z =8
3x − y +4z =7
x +3y +2z =−3
⎧
⎨
⎩
5x −3y +2z =−5
2x − y − z =4
3x −2y +2z =−7
⎧
⎨
⎩
3x −5y +4z =5
5x +2y + z =0
2x +3y −2z =3
⎧
⎨
⎩
4x −3y + z =7
2x −5y −4z =3
3x −2y −2z =−7
⎧
⎨
⎩
3x +8y +2z =−5
2x +5y −3z =0
x +2y −2z =−1
⎧
⎨
⎩
11x +9y +2z =−9
7x +5y +3z =−7
4x +3y + z =−3
⎧1
⎪3 x − y − z =1
⎨x + 5 2
⎪
y + z =−2
⎩2x +2y + 1 2 z =−4
⎧x + 1
⎪ 2 y + 1 2 z =0
⎨1
5 x − 1 5 y + z =0
⎪
1
⎩3 x − 1 3 y +2z =−1
⎧x + 1 y −2z =−1
⎪ 3
⎨1
3
⎪
x + y + 1 2 z =0
1
⎩2 x + 1 3 y − 1 2 z =−1
⎧1
⎪3 x − y + 1 2 z =4
⎨2
3
⎪
x + 5 y −4z =0
2
⎩x − 1 2 y + 3 2 z =2
⎧x +2z =0
⎨4y +3z =−2
⎩2x −5y =3
⎧2x +5y =4
⎨3y − z =3
⎩4x +3z =−3
⎧2y +3z =−1
⎨5x +3y =−6
⎩7x + z =1
⎧3x − z =−3
⎨5y +2z =−6
⎩4x +3y =−8
⎧
⎨
⎩
4x −3y +2z =0
−2x +3y −7z =1
2x −2y +3z =6
⎧
⎨
⎩
x −2y +2z =1
−2x + y − z =2
x − y + z =5

⎧
⎨
⎩
2x +3y + z =12
x + y + z =9
3x +4y +2z =20
⎧
⎨
⎩
x +4y + z =−8
4x − y +3z =9
2x +7y + z =0
⎧
⎨
⎩
x +2y + z =4
x + y −2z =3
−2x −3y + z =−7
⎧
⎨
⎩
x + y −2z =3
−2x −3y + z =−7
x +2y + z =4
⎧
⎨
⎩
x + y −3z =−1
y − z =0
−x +2y =1
⎧
⎨
⎩
x −2y +3z =1
x + y −3z =7
3x −4y +5z =7
a
b

3(x + 2) + 4 = 4(2x − 1) + 9.
0.25p + 0.25(p + 4) = 5.20.
x = −2 y = 3: 2x 2 − xy
+ 3y 2 .
m × n.
2×3.

a ⎧ −3y =−1
⎨5x b
⎩y =2x −2
⎧
⎨
⎩
6x −5y +2z =3
2x + y −4z =5
3x −3y + z =−1
a
y = 2x −2
−2x + y = −2
b
a ⎧ ⎨⎧
⎩
⎧
3x + 8y =−3
2x = −5y −3 b ⎨
⎩
2x − 5y + 3z = 8
3x − y + 4z = 7
⎩ x + 3y + 2z = −3
a ⎧ ⎨⎧
⎩
11x =−9y − 5
7x + 5y = −1
⎧
b ⎨
⎩
5x − 3y + 2z = −5
2x − y − z = 4
3x − 2y + 2z = −7
⎡
⎢ 4 −3 3 −1⎤
1 2 −1 2⎥.
⎣−2 −1 3 −4⎦
|
4×3

⎡1 −1 2 3
⎤
⎢2 1 −2 1⎥
⎣4 −1 2 0
⎦ .
⎡1 1 1 4
⎤
⎢2 3 −1 8⎥
⎣1 1 −1 3
⎦ .
−3
−3

a
b
c −2
a
⎡
⎢ 6 −5 2
|
2 1 −4
⎣3 −3 1
3⎤
5⎥
−1⎦
b
c −2
a
b
c
⎡ 5 −2 −2
⎢ 4 −1 −4
⎣−2 3 0
−2
⎤
4⎥
−1
⎦
|
a
b
c
⎡2 −3 −2
⎢4 1 −3
⎣5 0 4
−4
⎤
2⎥
−1
⎦
|

⎡
⎣ 1 −1 2⎤
4 −8 | 0⎦ .
−4
⎡1
−1 2
⎤
⎣3 −6 | 2⎦
.
⎡ 1 −1 3
⎤
⎣−2 −3 | 2⎦
.
⎧3x +4y =5
⎨
⎩x +2y =1 .

⎧2x + y = 7
⎨
⎩x
− 2y = 6 .
⎧2x + y = −4
⎨
⎩x
− y = −2 .

⎧
⎨
⎩
3x +8y +2z =−5
2x +5y −3z =0 .
x +2y −2z =−1

⎧
⎨
⎩
2x − 5y + 3z = 8
3x − y + 4z = 7 .
⎩ x + 3y + 2z = −3
⎧
⎨
⎩
−3x + y + z = −4
−x + 2y − 2z = 1.
2x − y − z = −1

⎧x + y +3z =0
⎨x +3y +5z =0.
⎩2x +4z =1
0≠1
⎧x
− 2y + 2z = 1
⎨−2x + y − z = 2.
⎩x
− y + z = 5
⎧
⎨
⎩
3x + 4y − 3z = −2
2x + 3y − z =−12.
⎩ x + y − 2z = 6

⎧
⎨
⎩
x −2y +3z =1
x + y −3z =7 .
3x −4y +5z =7
−2
0=0
y =2z +2.
(x, y, z), x = z +5;y =2z +2;z

⎧ x + y − z = 0
⎨2x + 4y − 2z = 6.
⎩3x + 6y − 3z = 9
⎧x
− y − z = 1
⎨−x + 2y − 3z = −4.
⎩3x − 2y − 7z = 0

a ⎧ − y =−1
⎨3x a ⎧ +4y =−5
⎨2x a ⎧ − y =−4
⎨3x
⎩2y =2x +5
⎩3x −2y =2
⎩2x = y +2
⎧4x +3y =−2
⎧3x −2y − z =−2
⎧x −3y −4z =−2
b ⎨x −2y −3z =7
b ⎨−2x + y =5
b ⎨4x +2y +2z =5
⎩2x − y +2z =−6
⎩5x +4y + z =−1
⎩2x −5y +7z =−8
a ⎧ −5y =−3
⎨2x
⎩4x =3y −1
⎧
b ⎨
⎩
4x +3y −2z =−3
−2x + y −3z =4
−x −4y +5z =−2
⎡ ⎣ 2 −1 4⎤
1 −3 | 2⎦ ⎡ ⎣ 2 −4 −2⎤
3 −3 | −1⎦
|
⎡1 0 −3
|
⎢1 −2 0
⎣0 −1 2
−1⎤
−2⎥
3⎦
⎡2 −2 0
⎢0 2 −1
⎣3 0 −1
−1⎤
2⎥
−2⎦
⎡ ⎣ 6 −4 3⎤
3 −2 | 1⎦
⎡ ⎣ 4 −6 −3⎤
⎡4 −12 −8
3 2 |
|
16⎤
1⎦
⎢4 −2 −3 −1⎥
⎣−6 2 −1 −1⎦
a a
a
b b
b
c −2 c
c −2
⎡
⎢ 6 −5 2
|
2 1 −4
⎣3 −3 1
3⎤
5⎥
−1⎦
a
b
c −2
⎡
⎣ 1 2 5⎤
−3 −4 | −1⎦ ⎡1 −2 3
|
−4⎤
⎢3 −1 −2 5⎥.
⎣2 −3 −4 −1⎦

⎧2x + y =2
⎧3x + y =2
⎨ ⎨
⎩x − y =−2
⎩x − y =2
⎧
⎨
⎩
−x +2y =−2
x + y =−4
⎧
⎨
⎩
−2x +3y =3
x +3y =12
⎧
⎨
⎩
2x −3y + z =19
−3x + y −2z = −15
x + y + z =0
⎧
⎨
⎩
2x − y +3z =−3
−x +2y − z =10
x + y + z =5
⎧
⎨
⎩
2x −6y + z =3
3x +2y −3z =2
2x +3y −2z =3
⎧
⎨
⎩
4x −3y + z =7
2x −5y −4z =3
3x −2y −2z =−7
⎧x +2z =0
⎨4y +3z =−2
⎩2x −5y =3
⎧2x +5y =4
⎨3y − z =3
⎩4x +3z =−3
⎧2y +3z =−1
⎨5x +3y =−6
⎩7x + z =1
⎧3x − z =−3
⎨5y +2z =−6
⎩4x +3y =−8
⎧
⎨
⎩
2x +3y + z =12
x + y + z =9
3x +4y +2z =20
⎧
⎨
⎩
x +2y +6z =5
−x + y −2z =3
x −4y −2z =1
⎧
⎨
⎩
x +2y −3z =−1
x −3y + z =1
2x − y −2z =2
⎧
⎨
⎩
4x −3y +2z =0
−2x +3y −7z =1
2x −2y +3z =6
⎧
⎨
⎩
x − y +2z =−4
2x + y +3z =2
−3x +3y −6z =12
⎧
⎨
⎩
−x −3y +2z =14
−x +2y −3z =−4
3x + y −2z =6
⎧
⎨
⎩
x + y −3z =−1
y − z =0
−x +2y =1
⎧
⎨
⎩
x +2y + z =4
x + y −2z =3
−2x −3y + z =−7
⎧ x + y =10
⎨
⎩x − y =6
a
b c
⎧
⎨
⎩
3x + y =12
x = y −8

a
b

2×2
3×3
5(−2) − (−4)(1).
−3(8 − 10) + (−2)(6 − 3) − 4 ⎛ ⎝−3 − (−4) ⎞ ⎠.
−12
−8 .
2×2
⎡ ⎣ a b ⎤
c d⎦ ,
|
a b
c
d| .
⎡ ⎣ a
c
a ⎡ ⎣ 4 −2 ⎤
3 −1⎦ b ⎡ ⎣ −3 −4 ⎤
−2 0⎦ .
a
b ⎤
d⎦ ,
|
a b
= ad − bc
c d|

a ⎡5
⎣ −3
2 −4⎦
⎤ b ⎡−4
⎣ −6
⎤
0 7⎦
.
a ⎡−1 ⎣ 3
−2 4⎦
⎤ b ⎡−7
⎣ −3
⎤
−5 0⎦
.
3×3
3×3
2×2 3×3
3×3
3×3 2×2
3×3
a 1 ,
2×2
b 2 ,
2×2

a
|
4
−2 3
1 0 −3
−2 −4 2|, a a 1 b b 3 c c 2 .
a 1 .
2×2
b
b 3 .
2×2
c
c 2 .

2×2
|
1
−1 4
0 2 −1
−2 −3 3|, a a 1 b b 2 c c 3 .
|−2 −1 0
3 0 −1 a a 2 b b 3 c c 2 .
−1 −2 3|,
3×3
3×3 2×2
3×3
3×3
3×3
|
2 −3 −1
3 2 0
−1 −1 −2|

|3 −2 4
0 −1 −2
2 3 −1|,
|
3 −2 −2
2 −1 4
−1 0 −3|,
3×3
|+ − +
+|
− + −
+ −
|+ − +
+|
− + −
+ −

|4 −1 3 0 2
5 −4 −3|
|2 −1 0 3 −4
3 −4 −3|
|−2 −1 −3
0|
−1 2 2
4 −4
⎧ a 1 x + b 1 y = k 1
⎨ , (x, y)
⎩a 2 x + b 2 y = k 2

D x D y ,
⎧2x + y =−4
⎨
⎩3x −2y =−6 .

⎧
⎨
⎩
3x + y = −3
2x + 3y = 6 .
⎧
⎨
⎩
−x + y = 2
2x + y = −4 .
D x .
D y .
x = D x
D ,
y = D y
D
3×3
⎧
⎨
⎩
a 1 x + b 1 y + c 1 z = k 1
a 2 x + b 2 y + c 2 z = k 2 , (x, y, z)
a 3 x + b 3 y + c 3 z = k 3

⎧
⎨
⎩
3x −5y +4z =5
5x +2y + z =0 .
2x +3y −2z =3
D x .
D y .

D z .
(2, −3, −4).
⎧
⎨
⎩
3x + 8y + 2z = −5
2x + 5y − 3z = 0 .
⎩ x + 2y − 2z = −1
⎧
⎨
⎩
3x + y − 6z = −3
2x + 6y + 3z = 0 .
3x + 2y − 3z = −6
D =0,
D =0 D x , D y D z
D =0 D x , D y D z
D =0,
Value of determinants Type of system Solution
D = 0 and D x , D y and D z are all zero consistent and dependent infinitely many solutions
D = 0 and D x , D y and D z are not all zero inconsistent no solution

⎧
⎨
⎩
x +3y =4
−2x −6y =3 .
Evaluate the determinant D, using the
coefficients of the variables.
⎧ x +3y =4
⎨
⎩−2x −6y =3
1 3
D =
| −2 −6|
D =−6−(−6)
D =0
D x D y ,
Evaluate the determinant D x . D x =
| 4 3
3 −6|
D x =−24−9
D x =15
⎧
⎨
⎩
4x − 3y = 8
8x − 6y = 14 .
⎧ x = −3y + 4
⎨
⎩2x + 6y = 8 .
(x 1 , y 1 ),
(x 3 , y 3 )
(x 2 , y 2 )
(x 1 , y 1 ),
|x 1 y 1
1|
1
x 2 y 2 1
x 3 y 3
=0
(x 2 , y 2 ) (x 3 , y 3 )
|x 1 y 1
1|
1
x 2 y 2 1
x 3 y 3
=0
(5, −5),
(4, −3), (3, −1)

(5, −5), (4, −3), (3, −1)
(3, −2), (5, −3), (1, −1)
(−4, −1), (−6, 2), (−2, −4)

⎡ ⎣ 6 −2 ⎤
3 −1⎦ ⎡ ⎣ −4 8 ⎤
−3 5⎦
⎡ ⎣ −3 5 ⎤
0 −4⎦
⎡ ⎣ −2 0 ⎤
7 −5⎦
|
3 −1
5|
4
|−1 −3 2
−1 0 −2
4 −2 −1
−4 1 −2 0 −3|
a a 1 b b 2 c c 3
|
2 −3 −4
−1 2 −3
0 −1 −2|
a a 2 b b 2 c c 2
a a 1 b b 1 c c 2
|−2 −2 3
1 −3 0
−2 3 −2|
a a 3 b b 3 c c 3
|−2 3 −1
−1 2 −2
3 1 −3| |
4 −1 −2
7|
−3 −2 1
−2 −5
|−2 −3 −4
0|
5 −6 7
−1 2
|1 3 −2
5 −6 4
0 −2 −1|
|−5 −1 −4
6| |
4 −1 3
4|
4 0 −3
3 −2 2
2 −2 −1 0
|
3 5 4
1|
|2 −4 −3
0|
−1 3 0
5 −1 −4
−2 6 3 2
⎧−2x +3y =3
⎧x −2y =−5
⎨ ⎨
⎩x +3y =12
⎩2x −3y =−4
⎧
⎨
⎩
x −3y =−9
2x +5y =4
⎧
⎨
⎩
2x + y =−4
3x −2y =−6
⎧
⎨
⎩
x −2y =−5
2x −3y =−4
⎧
⎨
⎩
x −3y =−9
2x +5y =4

⎧5x −3y =−1
⎨
⎩2x − y =2
⎧
⎨
⎩
3x +8y =−3
2x +5y =−3
⎧
⎨
⎩
6x −5y +2z =3
2x + y −4z =5
3x −3y + z =−1
⎧
⎨
⎩
4x −3y + z =7
2x −5y −4z =3
3x −2y −2z =−7
⎧
⎨
⎩
2x −5y +3z =8
3x − y +4z =7
x +3y +2z =−3
⎧
⎨
⎩
11x +9y +2z =−9
7x +5y +3z =−7
4x +3y + z =−3
⎧x +2z =0
⎨4y +3z =−2
⎩2x −5y =3
⎧2x +5y =4
⎨3y − z =3
⎩4x +3z =−3
⎧2y +3z =−1
⎨5x +3y =−6
⎩7x + z =1
⎧3x − z =−3
⎨5y +2z =−6
⎩4x +3y =−8
⎧
⎨
⎩
2x + y =3
6x +3y =9
⎧
⎨
⎩
x −4y =−1
−3x +12y =3
⎧
⎨
⎩
−3x − y =4
6x +2y = −16
⎧
⎨
⎩
4x +3y =2
20x +15y =5
⎧
⎨
⎩
x + y −3z =−1
y − z =0
−x +2y =1
⎧
⎨
⎩
2x +3y + z =12
x + y + z =9
3x +4y +2z =20
⎧
⎨
⎩
3x +4y −3z =−2
2x +3y − z = −12
x + y −2z =6
⎧
⎨
⎩
x −2y +3z =1
x + y −3z =7
3x −4y +5z =7
(0, 1), (2, 0), (−2, 2). (0, −5), (−2, −2), (2, −8).
(4, −3), (6, −4), (2, −2). (−2, 1), (−4, 4), (0, −2).
3×3
a
b

2a < 5a + 12.
⎛ ⎝ ⎛ 3, 1
2
⎞
⎠
⎞ y > 2x + 3.
⎧x +4y ≥10
⎨
⎩3x −2y

(−2, 4) (−2, 4)
b (3, 1)
(3, 1) (3, 1)
a (3, −1) b (6, −3)
⎧x
− 5y > 10
⎨
⎩2x + 3y > −2 .
a (−2, 1) b (4, −1)
⎧ y > 4x − 2
⎨
⎩4x − y < 20 .
⎧y ≥2x −1
⎨
⎩y < x +1 .

⎧ y < 3x + 2
⎨
⎩ y > −x − 1 .
⎧ y < − 1x + 3
⎨ 2 .
⎩ y < 3x − 4
⎧x − y >3
⎨
⎩y 3
⎨
⎩y

y −1 4 x + 5.
⎧
⎨
⎩
x −2y −4

⎧
⎨
⎩
x −2y −4
x −2y −4, y =−4
y =−4.
(0, 0)
⎧y
≥ 3x − 2
⎨
⎩ y < − 1 .
⎨⎧
⎧ x > −4
⎩x
− 2y ≥ −4 .

⎧4x +3y ≥12
⎨
⎩y

⎧3x − 2y ≥ 12
⎨
⎩y
≥ 3 2 x + 1 .
⎧ x + 3y > 8
⎨
⎩ y < − 1 3 x − 2.
⎧
⎨ y > 1 2 x −4 .
⎩x −2y 1 2 x −4
⎩x −2y 1 2 x −4 y = 1 2 x −4
m = 1 2

x −2y

x = the number of small photos.
y = the number of large photos
She wants to have at least 25 photos.
The number of small plus the number of large should be at least 25.
x + y ≥25
$4 for each small and $10 for each large must be no more than $200
4x +10y ≤ 200
The number of small photos must be greater than or equal to 0.
x ≥0
The number of large photos must be greater than or equal to 0.
y ≥0
⎧x + y ≥25
⎪
4x +10y ≤ 200
⎨
⎪x ≥0
⎩y ≥0
b
x ≥0 y ≥0
x + y ≥ 25, x + y =25
4x +10y ≤ 200, 4x +10y = 200
c
d

a
b
c
d
a
b
c
d
a
b
c
d
a

h = the number of hamburgers.
c = the number of cookies
240h + 160c ≥ 800
The amount spent on hamburgers at $1.40 each, plus the amount spent on cookies at $0.50 each
must be no more than $5.00.
1.40h + 0.50c ≤5
The number of hamburgers must be greater than or equal to 0.
h ≥0
The number of cookies must be greater than or equal to 0.
c ≥0
⎧240h + 160c ≥ 800
⎪
1.40h + 0.50c ≤5
⎨
⎪h ≥0
⎩c ≥0
b
h > = 0 c > = 0
240h + 160c ≥ 800,
240h + 160c = 800
1.40h + 0.50c ≤5. 1.40h + 0.50c =5.
c
d

a
b
c
d
a
b
c
d

⎧3x + y >5
⎧4x − y −8
⎨ y > 2 3 x −5
x + 1 2 y ≤4
a (3, −3) b (7, 1)
a (5, −2) b (−1, 3)
⎧
⎩
a (6, −4) b (3, 0)
⎧
⎨ y < 3 2 x +3
3
⎩ x −2y 14
⎨
⎩5x − y ≤8
a (2, 3) b (7, −1)
⎧6x −5y −8
a (1, −3) b (−4, 4)
⎧y ≤3x +2
⎧y x −1
⎩y ≥−x −1
⎧y 2x −3
⎧ x − y >1
⎨
⎩y −6
−x +2y ≥−4
⎧
⎨
⎩
x −2y 4
y ≤−1
⎧
⎨ y ≥−1 2 x −3
⎩x ≤2
⎧
⎨ y ≤−2 3 x +5
⎩x ≥3
⎧
⎨ y ≥ 3 4 x −2
⎩y 1 3 x +1

⎧−2x +6y 2x +4
⎧
⎨
⎩
−3x +6y >12
4y ≤2x −4
⎧
⎨
⎩
y ≥−3x +2
3x + y >5
⎧
⎨ y ≥ 1 2 x −1
⎩−2x +4y ≥4
⎧
⎨ y ≤−1 4 x −2
⎩x +4y −4
⎧
⎨
⎩
3y > x +2
−2x +6y >8
⎧
⎨ y < 3 4 x −2
⎩−3x +4y

a
b
c
d
a
b
c
d
x − y ≥3.
x − y =3
⎧x +2y ≤6
⎨
⎩y ≥− 1
x −4.
2
a
b

C(x) = R(x).
x
3×3 3×3 2×2
3×3
(x, y).
(x, y, z).

Choose the Most Convenient Method to Solve a System of Linear Equations
—————
Graphing
——————— Substitution
———————
Elimination
Use when you need a
picture of the situation.
Use when one equation is
already solved or can be
easily solved for one
variable.
Use when the equations are
in standard form.
C(x) = ⎛ ⎝cost per unit ⎞ ⎠ · x + fixed costs
R(x) = ⎛ ⎝sellingpriceperunit ⎞ ⎠ · x
C(x) = R(x)
ax + by + cz = d
(x, y, z)

m × n. 2×3.
⎡ ⎣ a b ⎤
c d⎦ ,
|
a b
= ad − bc
c d|
3×3

|+ − +
+|
− + −
+ −
⎧ a 1 x + b 1 y = k 1
⎨ , (x, y)
⎩a 2 x + b 2 y = k 2
D x .
D y .
x = D x
D ,
y = D y
D .
D =0,
Value of determinants Type of system Solution
D = 0 and D x , D y and D z are all zero consistent and dependent infinitely many solutions
D = 0 and D x , D y and D z are not all zero inconsistent no solution
(x 1 , y 1 ),
(x 2 , y 2 ), (x 3 , y 3 )
|x 1 y 1
1|
1
x 2 y 2 1
x 3 y 3
=0

⎧x +3y =−9
⎧ x + y =8
⎨ ⎨
⎩2x −4y =12
⎩y = x −4
a (−3, −2)
a (6, 2)
b (0, −3)
b (9, −1)
⎧
⎨
⎩
3x + y =6
x +3y =−6
⎧
⎨
⎩
x +4y =−1
x =3
⎧
⎨
⎩
2x − y =5
4x −2y =10
⎧−x +2y =4
⎨
⎩y = 1 2 x −3
⎧
⎨ y = 2 5 x +2
⎩−2x +5y =10
⎧
⎨
⎩
3x +2y =6
y =−3x +4
⎧5x −4y =0
⎨
⎩y = 5 4 x −5
⎧3x −2y =2
⎧ x − y =0
⎨
⎩y = 1 ⎨
2 x +3
⎩2x +5y = −14
⎧y =−2x +7
⎨
⎩y = 2 3 x −1
⎧ y =−5x ⎨
⎩5x + y =6
⎧
⎨ y =−1 3 x +2
⎩x +3y =6
⎧ x + y =12
⎨
⎩x − y = −10
⎧
⎨
⎩
3x −8y =20
x +3y =1
⎧
⎨
⎩
9x +4y =2
5x +3y =5
⎧
⎨
⎩
1
3 x − 1 2 y =1
3
4 x − y = 5 2
⎧
⎨
⎩
−x +3y =8
2x −6y = −20
⎧6x −5y =27
⎧y =3x −9
⎨ ⎨
⎩3x +10y = −24
⎩4x −5y =23

( 3
4

⎧
⎨
⎩
a (2,3,−1)
b (3,1,3)
3x −4y −3z =2
2x −6y + z =3
2x +3y −2z =3
⎧
y = 2 ⎪ 3 x −2
⎨
⎪
x +3y − z =15
⎩x −3y + z =−2
a ⎛ ⎝ −6, 5, 1 ⎞
2⎠
b ⎛ ⎝ 5, 4 3 ,−3⎞ ⎠
⎧
⎨
⎩
3x −5y +4z =5
5x +2y + z =0
2x +3y −2z =3
⎧x + 5 ⎪ 2 y + z =−2
⎨2x +2y + 1
2
⎪
z =−4
1
⎩3 x − y − z =1
⎧5x +3y =−6
⎨2y +3z =−1
⎩7x + z =1
⎧
⎨
⎩
2x +3y + z =12
x + y + z =9
3x +4y +2z =20
⎧
⎨
⎩
−x −3y +2z =14
−x +2y −3z =−4
3x + y −2z =6

⎧3x − y =−1
⎧4x +3y =−2
⎨
⎩−2x +2y =5
⎨x −2y −3z =7
⎩2x − y +2z =−6
⎡
⎣ 2 −4 −2⎤
⎡1 0 −3 −1⎤
3 −3 | −1⎦
⎢1 −2 0 −2⎥
⎣0 −1 2 3⎦
⎡
⎣ 4 −6 −3⎤
⎡1 −3 −2
3 2 |
|
4⎤
1⎦
⎢2 2 −1 −3⎥
⎣4 −2 −3 −1⎦
a
b
c
a
b
c −2
|
⎧
⎨
⎩
4x + y =6
x − y =4
⎧
⎨
⎩
2x − y +3z =−3
−x +2y − z =10
x + y + z =5
⎧2y +3z =−1
⎨5x +3y =−6
⎩7x + z =1
⎧
⎨
⎩
x +2y −3z =−1
x −3y + z =1
2x − y −2z =2
⎧
⎨
⎩
x + y −3z =−1
y − z =0
−x +2y =1
⎡
⎣ 8 −4 ⎤
5 −3⎦

|−1 −3 2
4 −2 −1
−2 0 −3|;
a a 1 b b 1 c c 2
|−2 −3 −4
0|
5 −6 7
−1 2
|
3 5 4
1|
−1 3 0
−2 6
⎧
⎨
⎩
x −3y =−9
2x +5y =4
⎧
⎨
⎩
4x −3y + z =7
2x −5y −4z =3
3x −2y −2z =−7
⎧2x +5y =4
⎨3y − z =3
⎩4x +3z =−3
⎧
⎨
⎩
x + y −3z =−1
y − z =0
−x +2y =1
⎧
⎨
⎩
3x +4y −3z =−2
2x +3y − z = −12
x + y −2z =6
(0, 2), (−1, −1),
(−2, 4)
⎧
⎨
⎩
a (2, −1)
b (3, −2)
4x + y >6
3x − y ≤12
⎧
⎨ y > 1 3 x +2
x − 1 4 y ≤10
⎩
a (6, 5)
b (15, 8)
⎧
⎨
⎩
y −1
⎨
⎩y < 1 3 x −2
⎧
⎨
⎩
2x −3y

⎧
⎨ y ≤−3 4 x +1
⎩x ≥−5
⎧x +3y −4
a
b
c
d
a
b
c
d

⎧ x − y =5
⎧ x − y >−2
⎨ ⎨
⎩x +2y =−4
⎩y ≤3x +1
⎧
⎨
⎩
x +4y =6
−2x + y =−3
⎧
⎨
⎩
−3x +4y =25
x −5y = −23
⎧
⎨
⎩
x + y − z =−1
2x − y +2z =8
−3x +2y + z =−9
⎧
⎨
⎩
2x + y =7
x −2y =6
⎧
⎨
⎩
−3x + y + z =−4
−x +2y −2z =1
2x − y − z =−1
⎧
⎨
⎩
3x + y =−3
2x +3y =6
|
3 −2 −2
2 −1 4
−1 0 −3|.
a
b
c
d
a
b
c

3x 2 + 3x + 1 + 8x 2 + 5x + 5.
(5n + 8) − (2n − 1).

4xyx 2 x = −2 y = 5.
ax m ,
−4a 2 b 3 c 2 .
ax m ,
y +1 4a 2 −7ab +2b 2 4x 4 + x 3 +8x 2 −9x +1
8y 2 −9x 3 y 5 −13a 3 b 2 c
a +7b 4x 2 − y 2 y 2 −16 3p 3 q −9p 2 q
x 2 −7x +12 9m 2 +2mn −8n 2 6k 4 − k 3 +8k z 4 +3z 2 −1
8ab 2
1+2,

a 7y 2 −5y +3 b −2a 4 b 2 c 3x 5 −4x 3 −6x 2 + x −8 d 2y −8xy 3 e
a 7y 2 −5y +3
b −2a 4 b 2
c 3x 5 −4x 3 −6x 2 + x −8
d 2y −8xy 3
e
a −5 b 8y 3 − 7y 2 − y − 3 c −3x 2 y − 5xyx + 9xyx 3 d 81m 2 − 4n 2 e −3x 6 y 3 z

a 64k 3 − 8 b 9m 3 + 4m 2 − 2 c 5 6 d 8a4 − 7a 3 b − 6a 2 b 2 − 4ab 3 + 7b 4 e −p 4 q 3
a 25y 2 +15y 2 b 16pq 3 − ⎛ ⎝−7pq 3⎞ ⎠.
a
25y 2 +15y 2
Combine like terms. 40y 2
b
16pq 3 − ⎛ ⎝−7pq 3⎞ ⎠
Combine like terms. 23pq 3
a 12q 2 + 9q 2 b 8mn 3 − ⎛ ⎝ ⎛ −5mn 3⎞ ⎠ ⎞ .
a −15c 2 + 8c 2 b −15y 2 z 3 − ⎛ ⎝ ⎛ −5y 2 z 3⎞ ⎠ ⎞ .
a a 2 +7b 2 −6a 2 b u 2 v +5u 2 −3v 2 .
a
a 2 +7b 2 −6a 2
−5a 2 +7b 2
b
u 2 v +5u 2 −3v 2
u 2 v +5u 2 −3v 2

a 8y 2 + 3z 2 − 3y 2 b m 2 n 2 − 8m 2 + 4n 2 .
a 3m 2 + n 2 − 7m 2 b pq 2 − 6p − 5q 2 .
⎛ ⎝7y 2 −2y +9 ⎞ ⎠ + ⎛ ⎝4y 2 −8y −7 ⎞ ⎠.
⎛
⎝ ____ 7y 2 −2y ___ +9 ⎞ ⎠ + ⎛ ⎝ ____ 4y 2 −8y ___ −7 ⎞ ⎠
_________ 7y 2 +4y 2
−2y −8y _______ +9−7
11y 2 −10y +2
⎛ ⎝ 7x 2 − 4x + 5 ⎞ ⎠ + ⎛
⎝x
2 − 7x + 3 ⎞ ⎠ .
⎛ ⎝ ⎛ 14y 2 + 6y − 4 ⎞ ⎠ ⎞ + ⎛ ⎝ ⎛ 3y 2 + 8y + 5 ⎞ ⎠ ⎞ .
⎛ ⎝9w 2 −7w +5 ⎞ ⎠ − ⎛ ⎝2w 2 −4 ⎞ ⎠.
⎛
⎝9w 2 −7w +5 ⎞ ⎠ − ⎛ ⎝2w 2 −4 ⎞ ⎠
____ 9w 2 −7w ___ +5− ____ 2w 2 +4
__________ 9w 2 −2w 2 −7w ___ +5+4
7w 2 −7w +9
⎛ ⎝ ⎛ 8x 2 + 3x − 19 ⎞ ⎠ ⎞ − ⎛ ⎝ ⎛ 7x 2 − 14 ⎞ ⎠ ⎞ .
⎛ ⎝ ⎛ 9b 2 − 5b − 4 ⎞ ⎠ ⎞ − ⎛ ⎝ ⎛ 3b 2 − 5b − 7 ⎞ ⎠ ⎞ .
a b, b − a, b

⎛ ⎝p 2 +10pq −2q 2⎞ ⎠ ⎛ ⎝p 2 + q 2⎞ ⎠.
⎛
⎝p 2 + q 2⎞ ⎠ − ⎛ ⎝p 2 +10pq −2q 2⎞ ⎠
p 2 + q 2 − p 2 −10pq +2q 2
p 2 − p 2 −10pq + q 2 +2q 2
−10pq +3q 2
⎛ ⎝ ⎛ a 2 + 5ab − 6b 2⎞ ⎠ ⎞
⎛
⎝ ⎛ a 2 + b 2⎞ ⎠ ⎞ .
⎛ ⎝ ⎛ m 2 − 7mn − 3n 2⎞ ⎠ ⎞
⎛
⎝ ⎛ m 2 + n 2⎞ ⎠ ⎞ .
⎛ ⎝u 2 −6uv +5v 2⎞ ⎠ + ⎛ ⎝3u 2 +2uv ⎞ ⎠.
⎛
⎝u 2 −6uv +5v 2⎞ ⎠ + ⎛ ⎝3u 2 +2uv ⎞ ⎠
u 2 −6uv +5v 2 +3u 2 +2uv
u 2 +3u 2 −6uv +2uv +5v 2
4u 2 −4uv +5v 2
⎛ ⎝ 3x 2 − 4xyx + 5y 2⎞ ⎠ ⎞ + ⎛ ⎝ 2x 2 − xy
⎞ ⎠ .
⎛ ⎝ 2x 2 − 3xyx − 2y 2⎞ ⎠ ⎞ + ⎛ ⎝ 5x 2 − 3xyx ⎞ ⎠ .
⎛ ⎝a 3 − a 2 b ⎞ ⎠ − ⎛ ⎝ab 2 + b 3⎞ ⎠ + ⎛ ⎝a 2 b + ab 2⎞ ⎠.

⎛
⎝a 3 − a 2 b ⎞ ⎠ − ⎛ ⎝ab 2 + b 3⎞ ⎠ + ⎛ ⎝a 2 b + ab 2⎞ ⎠
a 3 − a 2 b − ab 2 − b 3 + a 2 b + ab 2
a 3 − a 2 b + a 2 b − ab 2 + ab 2 − b 3
a 3 − b 3
⎛
⎝x
3 − x 2 y ⎞ ⎠ − ⎝
⎛ xy
2 + y 3⎞ ⎠ ⎞ + ⎛
⎝x
2 y + xy
2⎞ ⎠ ⎞ .
⎛
⎝p
3 − p 2 q ⎞ ⎠ + ⎛ ⎝p
⎛ q 2 + q 3⎞ ⎠ ⎞ − ⎝ ⎛
p
2 q + pq 2⎞ ⎠ ⎞ .
⎛ ⎝
f (x) = x 2 +5x +6 g(x) =3x −4
x 2 +5x +6 3x −4
f (x)
f (x) =5x 2 −8x +4 a f (4) b f (−2) c f (0).
a
b

c
f (x) = 3x 2 + 2x − 15, a f (3) b f (−5) c f (0).
g(x) = 5x 2 − x − 4, a g(−2) b g(−1) c g(0).
h(t) = −16t 2 + 250
t =2
h(t) = −16t 2 + 250
h(2), t =2. h(2) = −16(2) 2 + 250
h(2) = −16 · 4 + 250
h(2) = −64 + 250
h(2) = 186
h(t) = −16t 2 + 150
t = 0

h(t) = −16t 2 + 175
t = 3
f (x) g(x),
⎛
⎝f + g ⎞ ⎠(x) = f (x) + g(x)
⎛
⎝f − g ⎞ ⎠(x) = f (x) − g(x)
f (x) =3x 2 −5x +7 g(x) = x 2 −4x −3,
a ⎛ ⎝f + g ⎞ ⎠(x) b ⎛ ⎝f + g ⎞ ⎠(3) c ⎛ ⎝f − g ⎞ ⎠(x) d ⎛ ⎝f − g ⎞ ⎠(−2).
a
b ⎛ ⎝f + g ⎞ ⎠(x) ⎛ ⎝f + g ⎞ ⎠(3).
⎛
⎝f + g ⎞ ⎠(x) =4x 2 −9x +4
⎛ ⎝f + g ⎞ ⎠(3), x =3.
⎛
⎝f + g ⎞ ⎠(3) =4(3) 2 −9·3+4
⎛
⎝f + g ⎞ ⎠(3) =4·9−9·3+4
⎛
⎝f + g ⎞ ⎠(3) =36−27+4
⎛ ⎝f + g ⎞ ⎠(3) f (3) g(3)

f (3).
g(3).
⎛ ⎝f + g ⎞ ⎠(3).
c
d
f (x) = 2x 2 − 4x + 3 g(x) = x 2 − 2x − 6,
⎛
⎝f
− g ⎞ ⎠ ⎞ (−2).
a ⎛
⎝f
+ g ⎞ ⎠(x) b ⎛
⎝f
+ g ⎞ ⎠(3) c ⎛
⎝f
− g ⎞ ⎠(x) d
f (x) = 5x 2 − 4x − 1 g(x) = x 2 + 3x + 8,
⎛
⎝f
− g ⎞ ⎠ ⎞ (−2).
a ⎛
⎝f
+ g ⎞ ⎠(x) b ⎛
⎝f
+ g ⎞ ⎠(3) c ⎛
⎝f
− g ⎞ ⎠(x) d

a 47x 5 −17x 2 y 3 + y 2
b 5c 3 +11c 2 − c −8
c 5 9 ab + 1 3 b
d
e 4pq +17
a x 2 − y 2
b −13c 4
c a 2 +2ab −7b 2
d 4x 2 y 2 −3xy +8
e
a 8y −5x
b y 2 −5yz −6z 2
c y 3 −8y 2 +2y −16
d 81ab 4 −24a 2 b 2 +3b
e −18
a 11y 2
b −73
c 6x 2 −3xy +4x −2y + y 2
d 4y 2 +17z 2
e 5c 3 +11c 2 − c −8
a 5a 2 +12ab −7b 2
b 18xy 2 z
c 5x +2
d y 3 −8y 2 +2y −16
e −24
a 9y 3 −10y 2 +2y −6
b −12p 3 q
c a 2 +9ab +18b 2
d 20x 2 y 2 −10a 2 b 2 +30
e
a 14s −29t
b z 2 −5z −6
c y 3 −8y 2 z +2yz 2 −16z 3
d 23ab 2 −14
e −3
a 15xy
b
c 6x 2 −3xy +4x −2y + y 2
d 10p −9q
e m 4 +4m 3 +6m 2 +4m +1
a 7x 2 +5x 2
b 4a−9a
a 4y 3 +6y 3
b −y −5y
a −12w +18w
b 7x 2 y − (−12x 2 y)
a −3m +9m
b 15yz 2 −(−8yz 2 )
7x 2 +5x 2 +4a−9a
4y 3 +6y 3 − y −5y
−12w +18w +7x 2 y − (−12x 2 y)
−3m +9m +15yz 2 −(−8yz 2 )
a −5b −17b
b 3xy −(−8xy)+5xy

a −10x −35x
b 17mn 2 −(−9mn 2 )+3mn 2
a 2a 2 + b 2 −6a 2
b x 2 y −3x +7xy 2
a 12a +5b −22a
b pq 2 −4p −3q 2
a 5u 2 +4v 2 −6u 2
b 12a+8b
a 14x − 3y −13x
b a 2 b −4a −5ab 2
a xy 2 −5x −5y 2
b 19y + 5z
12a +5b −22a + pq 2 −4p −3q 2 14x − 3y −13x + a 2 b −4a −5ab 2 2a 2 + b 2 −6a 2 + x 2 y −3x +7xy 2
5u 2 +4v 2 −6u 2 + 12a+8b xy 2 −5x −5y 2 + 19y + 5z 4a, −3b, −8a
4x, 3y, −3x 5x 6 −12x 6 2p 4 −7p 4
(5y 2 +12y +4)+(6y 2 −8y +7)
(4y 2 +10y +3)+(8y 2 −6y +5)
(x 2 +6x +8)+(−4x 2 +11x −9)
(y 2 +9y +4)+(−2y 2 −5y −1)
(8x 2 −5x +2)+(3x 2 +3) (7x 2 −9x +2)+(6x 2 −4)
(5a 2 +8)+(a 2 −4a −9) (p 2 −6p − 18) + (2p 2 + 11)
(4m 2 −6m −3)−(2m 2 + m −7)
(3b 2 −4b +1)−(5b 2 − b −2)
(a 2 +8a +5)−(a 2 −3a +2)
(b 2 −7b +5)−(b 2 −2b +9)
(12s 2 −15s)−(s −9) (10r 2 −20r)−(r −8)
⎛ ⎝9x 2 +2 ⎞ ⎠ ⎛ ⎝12x 2 − x +6 ⎞ ⎠
⎛ ⎝5y 2 − y +12 ⎞ ⎠ ⎛ ⎝10y 2 −8y −20 ⎞ ⎠
⎛ ⎝7w 2 −4w +2 ⎞ ⎠ ⎛ ⎝8w 2 − w +6 ⎞ ⎠
⎛ ⎝5x 2 − x +12 ⎞ ⎠ ⎛ ⎝9x 2 −6x −20 ⎞ ⎠
⎛
⎝w 2 + w −42 ⎞ ⎠
⎛
⎝z 2 −3z −18 ⎞ ⎠
⎛
⎝w 2 −10w +24 ⎞ ⎠
⎛
⎝z 2 +5z −20 ⎞ ⎠

⎛ ⎝7x 2 −2xy +6y 2⎞ ⎠ + ⎛ ⎝3x 2 −5xy ⎞ ⎠
⎛ ⎝−5x 2 −4xy −3y 2⎞ ⎠ + ⎛ ⎝2x 2 −7xy ⎞ ⎠
⎛ ⎝7m 2 + mn −8n 2⎞ ⎠ + ⎛ ⎝3m 2 +2mn ⎞ ⎠
⎛ ⎝2r 2 −3rs −2s 2⎞ ⎠ + ⎛ ⎝5r 2 −3rs ⎞ ⎠
⎛ ⎝a 2 − b 2⎞ ⎠ − ⎛ ⎝a 2 +3ab −4b 2⎞ ⎠
⎛ ⎝m 2 +2n 2⎞ ⎠ − ⎛ ⎝m 2 −8mn − n 2⎞ ⎠
⎛ ⎝p 3 −3p 2 q ⎞ ⎠ + ⎛ ⎝2pq 2 +4q 3⎞ ⎠ − ⎛ ⎝3p 2 q + pq 2⎞ ⎠
⎛ ⎝a 3 −2a 2 b ⎞ ⎠ + ⎛ ⎝ab 2 + b 3⎞ ⎠ − ⎛ ⎝3a 2 b +4ab 2⎞ ⎠
⎛ ⎝x 3 − x 2 y ⎞ ⎠ − ⎛ ⎝4xy 2 − y 3⎞ ⎠ + ⎛ ⎝3x 2 y − xy 2⎞ ⎠
⎛ ⎝x 3 −2x 2 y ⎞ ⎠ − ⎛ ⎝xy 2 −3y 3⎞ ⎠ − ⎛ ⎝x 2 y −4xy 2⎞ ⎠
f (x) =8x 2 −3x +2,
a f (5) b f (−2) c f (0)
f (x) =5x 2 − x −7,
a f (−4) b f (1) c f (0)
g(x) =4−36x,
a g(3) b g(0) c g(−1)
g(x) =16−36x 2 ,
a g(−1) b g(0) c g(2)
h(t) = −16t 2 +75 h(t) = −16t 2 + 200
t =2
t =3
R(p) =−4p 2 + 420p.
p =60
R(p) =−4p 2 + 420p.
p =90
C(x) =6x 2 +90x
x =4
C(x) =6x 2 +90x
x =6
a b c d
f (x) =2x 2 −4x +1 g(x) =5x 2 +8x +3
f (x) =4x 2 −7x +3 g(x) =4x 2 +2x −1
f (x) =3x 3 − x 2 −2x +3 g(x) =3x 3 −7x
f (x) =5x 3 − x 2 +3x +4 g(x) =8x 3 −1

6y 2 +5y 4 11y 6 .
a
b

(−2)(−2)(−2).
8x
24y .
(−2.6)(4.21).
a m ,
m th
a m ,

x 2 · x 3 x 2+3 x 5 .
a m · a n = a m + n
a y 5 · y 6 b 2 x ·2 3x c 2a 7 ·3a.
a
a m · a n = a m + n .
b
a m · a n = a m + n .
c
a = a 1 .
a m · a n = a m + n .

d
a b 9 · b 8 b 4 2x · 4 x c 3p 5 · 4p d x 6 · x 4 · x 8 .
a x 12 · x 4 b 10 · 10 x c 2z · 6z 7 d b 5 · b 9 · b 5 .
x 5 x 2
x 2 x 3
x · x · x · x · x
x · x
x · x
x · x · x
x · x · x · x · x
x · x
x · x ·1
x · x · x
x 3 1 x
x5
x 2 x5−2 x 3 x2
x 3
1 x .
x
=1
x
a ≠0,
a m
a n = am − n , m > n and a m
a n = 1
a n − m,
n > m
a x9
x 7 b 310
c b8 d
73
3 2 12
b 7 5.
a

9>7,
x
a m
a n = am − n .
b
10 > 2,
3
a m
a n = am − n .
c
12 > 8,
b
a m
a n = 1
a n − m.
d
5>3,
3
a m
a n = 1
a n − m.
a x15 b
614
10
x
c x18 d
1215
6 5 22
x 12 30.
a y43 b
1015
37
y
c m7 d
98
10 7 15
m 9 19.
am
a m. , x =1, x (x ≠0)

am
a
m. m > n n < m
m = n? am m
a
a ≠0:
am
a m a0 a 0 =1.
a 0 =1.
a 9 0 b n 0 .
a
9 0
Use the definition of the zero exponent. 1
b
n 0
Use the definition of the zero exponent. 1
a 11 0 b q 0 .
a 23 0 b r 0 .
x2
x2
x
5.
x 5
x 2−5 x −3 .
x2
x 5

x −3 = 1 a ≠0,
3
x
a −n = 1 a n.
1
a −n
a −n = 1 a n. 1
1
a n
1·an
1
a n
1
a −n = an
a ≠0, a −n = 1 a n 1
a −n = an .
x −3 ,
1
x
3.
a x −5 b 10 −3 c 1
y −4 d 1
3 −2.
a
x −5
a −n = 1 a n. 1
x 5
b

10 −3
a −n = 1 a n. 1
10 3
1
1000
c
1
y −4
1
a −n = an . y 4
d
1
3 −2
1
a −n = an . 3 2
9
a z −3 b 10 −7 c 1
p −8 d 1
4 −3.
a n −2 b 10 −4 c 1
q −7 d 1
2 −4.

⎛
⎝ 3 −2
⎞
4⎠
a −n = 1 1
a n. 2
⎛
⎝ 3 ⎞
4⎠
1
9
16
16
9
16 9 ⎛ ⎝ 4 ⎞
3⎠2 .
⎛
⎝ 3 −2 2
⎞ ⎛
4⎠
= 4⎞ ⎝3⎠
a ≠0,b ≠0
a ⎛5 ⎞
⎝7⎠−2 b
⎛
⎝ − x y ⎞ ⎠−3
.
and ⎛ ⎝ a −n
⎞
b⎠
= ⎛b ⎝a ⎞ n
⎠
a
⎛ 5 −2
⎞
⎝7⎠
⎛ ⎝ a −n
⎞
b⎠
= ⎛b ⎝a ⎞ ⎠n
.
⎛
⎝ 7 2
⎞
5⎠
49
25
b

⎛
⎝ − x y ⎞ −3
⎠
⎛ ⎝ a −n
⎞
b⎠
= ⎛b ⎝a ⎞ ⎠n
.
⎛
⎝ − y x ⎞ 3
⎠
− y3
x 3
a ⎛2⎞ −4
⎝ 3⎠
b ⎛ ⎝ − m n ⎞−2 ⎠ .
a ⎛3⎞ −3
⎝ 5⎠
b ⎛ ⎝ − a⎞ −4
b⎠ .
a z −5 · z −3 b ⎛ ⎝m 4 n −3⎞ ⎠ ⎛ ⎝m −5 n −2⎞ ⎠ c ⎛ ⎝2x −6 y 8⎞ ⎠ ⎛ ⎝−5x 5 y −3⎞ ⎠.
a
z −5 · z −3
z −5 − 3
z −8
1
z 8
b
⎛
⎝m 4 n −3⎞ ⎠ ⎛ ⎝m −5 n −2⎞ ⎠
m 4 m −5 · n −2 n −3
m −1 · n −5
1
m 1 · 1
n 5
1
mn 5

c
⎛
⎝2x −6 y 8⎞ ⎠ ⎛ ⎝−5x 5 y −3⎞ ⎠
2(−5) · ⎛⎝x −6 x 5⎞ ⎠· ⎛⎝y 8 y −3⎞ ⎠
−10 · x −1 · y 5
a −n = 1 a n.
−10 · 1x · y5
−10y 5
x
a z −4 · z −5 b ⎛
⎝p
6 q −2⎞ ⎠ ⎞⎛ ⎝p
−9 q −1⎞ ⎞ ⎠ c ⎛ ⎝ 3u −5 v 7⎞ ⎠ ⎞⎛ ⎝ −4u 4 v −2⎞ ⎠ ⎞ .
a c −8 · c −7 b ⎛ ⎝ ⎛ r 5 s −3⎞ ⎠ ⎞⎛ ⎝ ⎛ r −7 s −5⎞ ⎠ ⎞ c ⎛ ⎝ ⎛ −6c −6 d 4⎞ ⎠ ⎞⎛ ⎝ ⎛ −5c −2 d −1⎞ ⎠ ⎞ .
(x 2 ) 3
What does this mean? x 2 · x 2 · x 2
(x 2 ) 3 x 2·3 x 6 .
(a m ) n = a m · n
a (y 5 ) 9 b (4 4 ) 7 c (y 3 ) 6 (y 5 ) 4 .

a
(a m ) n = a m · n .
b
c
(y 3 ) 6 (y 5 ) 4
y 18 · y 20
y 38
a (b 7 ) 5 b (5 4 ) 3 c (a 4 ) 5 (a 7 ) 4 .
a (z 6 ) 9 b (3 7 ) 7 c (q 4 ) 5 (q 3 ) 3 .
(2x) 3
2x ·2x ·2x
2·2·2·x · x · x
x 2 3 · x 3
(2x) 3 2 3 · x 3 .
(ab) m = a m b m

a (−3mn) 3 b ⎛ ⎝−4a 2 b ⎞ ⎠0
c
⎛
⎝ 6k 3⎞ ⎠−2
d
⎛
⎝ 5x −3⎞ ⎠2
.
a
(ab) m = a m b m .
b
⎛
⎝−4a 2 b ⎞ 0
⎠
(ab) m = a m b m . (−4) 0 ⎛ ⎝a 2⎞ ⎠0
(b)
0
1·1·1
1
c
⎛
⎝6k 3⎞ −2
⎠
(ab) m = a m b m .
(a m ) n = a m · n .
(6) −2 ⎛ ⎝k 3⎞ −2
⎠
6 −2 k −6
a −n = 1 a n. 1
6 2 · 1
k 6
1
36k 6
d

⎛
⎝5x −3⎞ 2
⎠
(ab) m = a m b m .
5 2 ⎛ ⎝x −3⎞ 2
⎠
25 · x −6
x −6 a −n = 1 a n. 25 · 1
x 6
25
x 6
a (2wx) 5 b ⎛ ⎝ ⎛ −11pq 3⎞ ⎠ ⎞0 c ⎛ ⎝ ⎛ 2b 3⎞ ⎠ ⎞−4 d ⎛ ⎝ ⎛ 8a −4⎞ ⎠ ⎞2 .
a (−3y) 3 b (−8m 2 n 3 ) 0 c ⎛ ⎝ ⎛ −4x 4⎞ ⎠ ⎞−2 d ⎛ ⎝ ⎛ 2c −4⎞ ⎠ ⎞3 .
⎛ x ⎝y ⎞ 3
⎠
x
y · xy · xy
x · x · x
y · y · y
x 3
y 3
⎛x ⎝y ⎞ 3
⎠ x3
y 3.
a b b ≠0, m
⎛
⎝ a m
⎞
b⎠
= am
b m

a ⎛ ⎝ b ⎞
3⎠
4
b
⎛
⎝
k
j
⎞
⎠
−3
c
⎛
3
⎜ 2xy2 ⎞
⎟
⎝
z d ⎛ 2
⎞
⎠ ⎝ ⎜4p−3
q 2 ⎟ .
⎠
a
⎛ ⎝ a m
⎞
b⎠
= am
b m.
b
c
3
⎛
⎜ 2xy2 ⎞
⎟
⎝
z
⎠
⎛ ⎝ a m
⎞
b⎠
= am
b m.
(ab) m = a m b m .
⎛
⎝ 2xy 2⎞ 3
⎠
z 3
8x 3 y 6
z 3
d

2
⎛
⎜ 4p−3 ⎞
⎝ q 2 ⎟
⎠
⎛ ⎝ a m
⎞
b⎠
= am
b m.
(ab) m = a m b m .
(a m ) n = a m · n .
⎛
⎝ 4p −3⎞ 2
⎠
⎛
⎝q 2⎞ 2
⎠
4 2 ⎛ ⎝p −3⎞ 2
⎠
⎛
⎝q 2⎞ 2
⎠
16p −6
q 4
16
q 4 · 1
p 6
16
p 6 q 4
a ⎛ p ⎞ 4
⎝10
⎠ b ⎛m ⎛ ⎞ −7 ⎝ n ⎠ c ⎛ 3ab
3⎞ 4
⎝ c 2 ⎠
d ⎛ ⎜⎛
⎝
3x −2
3
⎞
y 3 ⎟ .
⎠
a ⎛−2⎞ 3
⎝ q ⎠ b ⎛w ⎛ ⎞ −4 ⎝ x ⎠ c ⎛ ⎜⎛ xy
3 2
⎞
⎝3z 2 ⎟
⎠
d ⎛ 2m
−2⎞ 3
⎝ ⎠
.
n −2

a m · a n = a m + n
(a m ) n = a m · n
(ab) m = a m b m
a m
a n = am − n , a ≠0
a 0 =1,a ≠0
m
⎛a⎞ ⎝b⎠ = am
b
m, b ≠0
a −n = 1 a n 1 −n = an
a
−n
⎛a⎞ ⎝b⎠ = ⎛b ⎝a ⎞ n
⎠
a (3x 2 y) 4 (2xy 2 ) 3 b (x3 ) 4 (x −2 ) 5
(x 6 ) 5 c ⎛ ⎝ ⎜ 2xy2
⎞
x 3 y −2 ⎠
2 ⎛ ⎞
⎟
⎝ ⎜12xy3
x 3 y −1 ⎟
⎠
−1
.
a
(3x 2 y) 4 (2xy 2 ) 3
(ab) m = a m b m . (3 4 x 8 y 4 )(2 3 x 3 y 6 )
(81x 8 y 4 )(8x 3 y 6 )
81 · 8 · x 8 · x 3 · y 4 · y 6
648x 11 y 10
b

(x 3 ) 4 (x −2 ) 5
(x 6 ) 5
(a m ) n = a m · n .
(x 12 )(x −10 )
(x 30 )
x 2
x 30
a m
a n = 1
a n − m. 1
x 28
c
2 −1
⎛
⎜ 2xy2 ⎞ ⎛ ⎞
⎝x 3 y −2 ⎟
⎠ ⎝ ⎜12xy3
x 3 y −1 ⎟
⎠
⎛ ⎝ a m
⎞
b⎠
= am
b m.
(ab) m = a m b m .
⎛
⎜ 2y4 2
⎞ ⎛
⎝ x 2 ⎟
⎠ ⎝ ⎜ 12y4 −1
⎞
x 2 ⎟
⎠
⎛
⎝ 2y 4⎞ 2
⎛
⎠
⎛
⎝x 2⎞ 2
⎠
⎝12y 4⎞ −1
⎠
−1
⎛
⎝x 2⎞ ⎠
4y 8
x 4 · 12−1 y −4
x −2
4y 4
12x 2
y 4
3x 2
a (c 4 d 2 ) 5 (3cd 5 ) 4 b (a−2 ) 3 (a 2 ) 4
c ⎛
(a 4 ) 5 ⎜⎛ 3xyx 2 2 ⎞ ⎛
⎝x 2 y −3 ⎟
⎠
⎜⎛ 9xyx −3 −1
⎞
⎝ x 3 y 2 ⎟ .
⎠
a (a 3 b 2 ) 6 (4ab 3 ) 4 b (p−3 ) 4 (p 5 ) 3
c ⎛
(p 7 ) 6 ⎜⎛ 4x 3 y 2 2 ⎞ ⎛
⎝x 2 y −1 ⎟
⎠
⎜⎛ 8xyx −3 −1
⎞
⎝ x 2 ⎟ .
y
⎠

4 × 1,000
4× 1
1,000 .
4 × 1,000 4×10 3
4× 1
1,000
4× 1
10 3 4×10 −3
a ×10 n where 1 ≤ a < 10 and n is an integer.
×
4,000 = 4 × 10 3
0.004 = 4 × 10 −3
10 n .
10 −n .
a b 0.0052.
a

3.7 × 10 4
3.7 × 10,000
37,000
b
Check:
5.2 × 10 −3
5.2 × 1
10 3
5.2 ×
1000
1
5.2 × 0.001
0.0052
a b
a b
9.12 × 10 4 9.12 × 10 −4
9.12 × 10,000 9.12 × 0.0001
91,200 0.000912

|n|
a 6.2 × 10 3 b −8.9 × 10 −2 .
a
b
−2.
a 1.3 × 10 3 b −1.2 × 10 −4 .
a −9.5 × 10 4 b 7.5 × 10 −2 .

a ⎛ ⎝−4 × 10 5⎞ ⎠ ⎛ ⎝2×10 −7⎞ ⎠ b 9×103
3×10 −2.
a
⎛
⎝−4 × 10 5⎞ ⎠ ⎛ ⎝2×10 −7⎞ ⎠
−4·2·10 5 ·10 −7
−8 × 10 −2
−0.08
b
9×10 3
9×10 −2
9
3 × 103
10 −2
3×10 5
300,000
a ⎛ ⎝ ⎛ −3 × 10 5⎞ ⎠ ⎞⎛ ⎝ ⎛ 2 × 10 −8⎞ ⎠ ⎞ b 8 × 102
4 × 10 −2.
a ⎛ ⎝ ⎛ −3 × 10 −2⎞ ⎠ ⎞⎛ ⎝ ⎛ 3 × 10 −1⎞ ⎠ ⎞ b 8 × 104
2 × 10 −1.

a d 3 · d 6 b 4 5x ·4 9x c 2y ·4y 3 d w · w 2 · w 3 a x 4 · x 2 b 8 9x ·8 3 c 3z 25 ·5z 8 d y · y 3 · y 5
a n 19 · n 12 b 3 x ·3 6 c 7w 5 ·8w d a 4 · a 3 · a 9 a q 27 · q 15 b 5 x ·5 4x c 9u 41 ·7u 53
d c 5 · c 11 · c 2
m x · m 3 n y · n 2
y a · y b
x p · x q
a x18
x 3 b 512
c q18 d 102
5 3 36
q 10 3 a y20 b
716
c t10 d
83
10
y 7 2 40
t 8 5
a p21
p 7 b 416
4 4 c b b 9 d 4 4 6 a u24
u 3 b 915
9 5 c x
x 7 d 10
10 3
a 20 0 b b 0 a 13 0 b k 0
a −27 0 b − ⎛ ⎝27 0⎞ ⎠
a −15 0 b − ⎛ ⎝15 0⎞ ⎠
a a −2 b 10 −3 c 1
c −5 d 1
3 −2 a b −4 b 10 −2 c 1
b −3 d 1
5 −2
a r −3 b 10 −5 c 1
q −10 d 1
10 −3 a s −8 b 10 −2 c 1
t −9 d 1
10 −4
a ⎛ ⎝ 5 ⎞
8⎠−2 b
⎛
⎝ − b a ⎞ −2
⎠
a ⎛ ⎝ 4 ⎞
9⎠−3 b
⎛
⎝ − u v ⎞ −5
⎠
a (−5) −2 b −5 −2 c ⎛ ⎝ −1 5
a 3·5 −1 b (3·5) −1
−2 −2
⎞
⎠ d
⎛ − 1⎞ ⎝5⎠ a ⎛ ⎝10
3 ⎞
⎠−2
b
⎛
⎝ − 2 z ⎞ −3
⎠
a ⎛ ⎝ 7 ⎞
2⎠−3 b
⎛
⎝ − 3 x ⎞ −3
⎠
a −5 −3 b ⎛ ⎝ −1 5
⎞
⎠
a 3·4 −2 b (3·4) −2
−3
c −
⎛
⎝
1
5
⎞
⎠
−3
d (−5)
−3

a b 4 b −8 b ⎛ ⎝w 4 x −5⎞ ⎠ ⎛ ⎝w −2 x −4⎞ ⎠ c
⎛
⎝−6c −3 d 9⎞ ⎠ ⎛ ⎝2c 4 d −5⎞ ⎠
a s 3 · s −7 b ⎛ ⎝m 3 n −3⎞ ⎠ ⎛ ⎝m −5 n −1⎞ ⎠ c
⎛
⎝−2j −5 k 8⎞ ⎠ ⎛ ⎝7j 2 k −3⎞ ⎠
a a 3 · a −3 b ⎛ ⎝uv −2⎞ ⎠ ⎛ ⎝u −5 v −3⎞ ⎠ c
⎛
⎝−4r −2 s −8⎞ ⎠ ⎛ ⎝9r 4 s 3⎞ ⎠
p 5 · p −2 · p −4
a y 5 · y −5 b ⎛ ⎝pq −4⎞ ⎠ ⎛ ⎝p −6 q −3⎞ ⎠ c
⎛
⎝−5m 4 n 6⎞ ⎠ ⎛ ⎝8m −5 n −3⎞ ⎠
x 4 · x −2 · x −3
−4
a (m 4 ) 2 b (10 3 ) 6 c ⎛ ⎝x 3⎞ ⎠
a (b 2 ) 7 b (3 8 ) 2 c ⎛ ⎝k 2⎞ ⎠
−5
a ⎛ ⎝y 3⎞ ⎠x
b (5 x ) y c ⎛ ⎝q 6⎞ ⎠
−8
a ⎛ ⎝x 2⎞ ⎠y
b (7 a ) b c ⎛ ⎝a 9⎞ ⎠
−10
2
2
a (−3xy) 2 b (6a) 0 c ⎛ ⎝5x 2⎞ ⎠−2
d
⎛
⎝ −4y −3⎞ ⎠ a (−4ab) 2 b (5x) 0 c ⎛ ⎝4y 3⎞ ⎠−3
d
⎛
⎝ −7y −3⎞ ⎠
a (−5ab) 3 b ⎛ ⎝−4pq ⎞ ⎠0
c ⎛ ⎝−6x 3⎞ ⎠−2
d
⎛
⎝ 3y −4⎞ ⎠
2
a ⎛ ⎝−3xyz ⎞ ⎠4
b (−7mn) 0 c ⎛ ⎝−3x 3⎞ ⎠
d ⎛ ⎝2y −5⎞ 2
⎠
−2
a ⎛ 5
3
−6
p⎞
⎛
⎝2⎠
b
⎛x⎞ ⎝y⎠ c ⎜ 2xy2 ⎞
⎟
⎝
z d ⎛ 2
⎞
⎠ ⎝ ⎜4p−3
q 2 ⎟ a ⎛ ⎝ x 4 −5
⎞
⎠
3⎠
b
⎛a⎞ ⎛
⎝b⎠ c ⎜ 2x2 y 3 2
⎞
⎝ z 2 ⎟
⎠
d ⎛ 2
⎝ ⎜x3 y⎞
z 4 ⎟
⎠
a ⎛ ⎝3b
a 4
⎞
⎠
⎛
⎜ p−1 q 4 2
⎞
⎝ r −4 ⎟
⎠
b ⎛ ⎝4m
5 −2
⎞
⎠
c ⎛ ⎝ 3a−2 b 3 −2
⎞
c 2 ⎠
d
a ⎛ ⎝2y
x
3
⎞
⎠
⎛
⎝ 5a3 b −1 −3
⎞
2c 4 ⎠
b ⎛ ⎝ 10
−4
⎞
3q⎠
c ⎛ ⎝ ⎜2x3 y 4
3z 2
5
⎞
⎟
⎠
d
a (5t 2 ) 3 (3t) 2 b
2 −1
⎛
⎜ 2xy2 ⎞ ⎛ ⎞
⎝x 3 y −2 ⎟
⎠ ⎝ ⎜12xy3
x 3 y −1 ⎟
⎠
⎛
⎝t 2⎞ ⎛⎝
⎠5
t −4⎞ 2
⎠
c
7
⎛
⎝t 3⎞ ⎠
a (10k 4 ) 3 (5k 6 ) 2 b
⎛
⎝q 3⎞ ⎛⎝
⎠6
q −2⎞ 3
⎠
⎛
⎝q 4⎞ 8
⎠
a (m 2 n) 2 (2mn 5 ) 4 b
⎛
⎝−2p −2⎞ ⎛⎝
⎠4
3p 4⎞ 2
⎠
⎛
⎝−6p 3⎞ 2
⎠
a (3pq 4 ) 2 (6p 6 q) 2 b
⎛
⎝−2k −3⎞ ⎛⎝
⎠2
6k 2⎞ 4
⎠
⎛
⎝9k 4⎞ 2
⎠

a 7n −1 b (7n) −1 c (−7n) −1
a 6r −1 b (6r) −1 c (−6r) −1
a ⎛ ⎝3p ⎞ ⎠−2
b 3p −2 c −3p −2 a ⎛ ⎝2q ⎞ −4
⎠ b 2q −4 c −2q −4
⎛ ⎝x 2⎞ ⎠4
· ⎛
⎝ x 3⎞ 2
⎠
⎛ ⎝y 4⎞ ⎠3
· ⎛
⎝ y 5⎞ 2
⎠
⎛ ⎝a 2⎞ ⎠6
· ⎛
⎝ a 3⎞ 8
⎠
⎛ ⎝b 7⎞ ⎠5
· ⎛
⎝ b 2⎞ 6
⎠
(2m 6 ) 3 (3y 2 ) 4
(10x 2 y) 3 (2mn 4 ) 5
(−2a 3 b 2 ) 4 (−10u 2 v 4 ) 3
⎛ ⎝ 2 3 x2 y ⎞ 3
⎠
⎛ ⎝ 7 9 pq4 ⎞
⎠
2
(8a 3 ) 2 (2a) 4 (5r 2 ) 3 (3r) 2
(10p 4 ) 3 (5p 6 ) 2 (4x 3 ) 3 (2x 5 ) 4
⎛ ⎝ 1 2 x2 y 3⎞ ⎛⎝
⎠4
4x 5 y 3⎞ 2
⎠
⎛ ⎝ 1 3 m3 n 2 4
⎞ ⎛⎝
⎠ 9m 8 n 3⎞ 2
⎠
(3m 2 n) 2 (2mn 5 ) 4 (2pq 4 ) 3 (5p 6 q) 2
a (3x) 2 (5x) b (2y) 3 (6y)
a ( 1 2 y2 ) 3 ( 2 3 y)2 b ( 1 2 j2 ) 5 ( 2 5 j3 ) 2
a (2r −2 ) 3 (4 −1 r) 2 b (3x −3 ) 3 (3 −1 x 5 ) 4 ⎛ ⎝ k−2 k 8
k 3 ⎞
⎠
2
⎛
⎜ j−2 j 5 3
⎞
⎝ j 4 ⎟
⎠
⎛
⎝−4m −3⎞ ⎛⎝
⎠2
5m 4⎞ ⎠
⎛
⎝−10m 6⎞ 3
⎠
3
⎛
⎝−10n −2⎞ ⎛⎝
⎠3
4n 5⎞ ⎠
2
⎛
⎝2n 8⎞ ⎠
2

a b
a b
a b
a b
a 5.2 × 10 2 b 2.5 × 10 −2
a −8.3 × 10 2 b 3.8 × 10 −2
a 7.5 × 10 6 b −4.13 × 10 −5
a 1.6 × 10 10 b 8.43 × 10 −6
a ⎛ ⎝3×10 −5⎞ ⎠ ⎛ ⎝3×10 9⎞ ⎠ b 7×10−3
1×10 −7 a ⎛ ⎝2×10 2⎞ ⎠ ⎛ ⎝1×10 −4⎞ ⎠ b 5×10−2
1×10 −10
a ⎛ ⎝7.1 × 10 −2⎞ ⎠ ⎛ ⎝2.4 × 10 −4⎞ ⎠ b 6×104
3×10 −2 a ⎛ ⎝3.5 × 10 −4⎞ ⎠ ⎛ ⎝1.6 × 10 −2⎞ ⎠ b 8×106
4×10 −1
x · x = x 2 .
a24
a 6 a4 .
−5 3 = (−5) 3 −5 4 ≠ (−5) 4 .
a
b

2(x + 3).
a 9 2 b (−9) 2 c −9 2 .
2x 2 − 5x + 3 x = −2.
a ⎛ ⎝3x 2⎞ ⎠ ⎛ ⎝−4x 3⎞ ⎠ b ⎛ ⎝ 5 6 x3 y ⎞ ⎠ ⎛ ⎝12xy 2⎞ ⎠.
a
⎛
⎝3x 2⎞ ⎠ ⎛ ⎝−4x 3⎞ ⎠
3·(−4) · x 2 · x 3
−12x 5
b

⎛
⎝ 5 6 x3 y ⎞ ⎠ ⎛ ⎝12xy 2⎞ ⎠
5
6 ·12·x3 · x · y · y 2
10x 4 y 3
a ⎛ ⎝ 5y 7⎞ ⎠ ⎞⎛ ⎝ −7y 4⎞ ⎞ ⎠ b ⎛ 2
⎝ 5
⎛ a 4 b 3 ⎞⎛ ⎠⎝ 15ab 3⎞ ⎠ ⎞ .
a ⎛ ⎝ −6b 4⎞⎛ ⎠ ⎝ −9b 5⎞ ⎞ ⎠ b ⎛ 2
⎝ 3
⎛ r 5 s ⎞ ⎠ ⎞⎛⎛ ⎝ 12r 6 s 7⎞ ⎠ ⎞ .
a −2y ⎛ ⎝4y 2 +3y −5 ⎞ ⎠ b 3x 3 y(x 2 −8xy + y 2 ).
a
b
3x 3 y(x 2 −8xy + y 2 )
3x 3 y · x 2 + ⎛ ⎝3x 3 y ⎞ ⎠·(−8xy)+ ⎛ ⎝3x 3 y ⎞ ⎠· y 2
3x 5 y −24x 4 y 2 +3x 3 y 3
a −3y ⎛ ⎝ 5y 2 + 8y − 7 ⎞ ⎠ b 4x 2 y 2 ⎛ ⎝ 3x 2 − 5xyx + 3y 2⎞ ⎠ ⎞ .
a 4x 2 (2x 2 − 3x + 5) b −6a 3 b(3a 2 − 2ab + 6b 2 ).

a ⎛ ⎝y +5 ⎞ ⎠ ⎛ ⎝y +8 ⎞ ⎠ b ⎛ ⎝4y +3 ⎞ ⎠ ⎛ ⎝2y −5 ⎞ ⎠.
a
(y +8).
b
a (x + 8)(x + 9) b (3c + 4)(5c − 2).
a (5x + 9)(4x + 3) b ⎛ ⎝ ⎛ 5y + 2 ⎞ ⎠ ⎞⎛ ⎝⎛ 6y − 3 ⎞ ⎠ ⎞ .
(x +3)(x +7)

a ⎛ ⎝y −7 ⎞ ⎠ ⎛ ⎝y +4 ⎞ ⎠ b (4x +3)(2x −5).
a
b

a (x − 7)(x + 5) b (3x + 7)(5x − 2).
a (b − 3)(b + 6) b ⎛ ⎝ ⎛ 4y + 5 ⎞ ⎠ ⎞⎛ ⎝⎛ 4y −10 ⎞ ⎠ ⎞ .
a ⎛ ⎝n 2 +4 ⎞ ⎠(n −1) b ⎛ ⎝3pq +5 ⎞ ⎠ ⎛ ⎝6pq −11 ⎞ ⎠.
a
b
a ⎛
⎝x
2 + 6 ⎞ ⎠ (x − 8) b (2ab + 5)(4ab − 4).

a ⎛
⎝y
2 + 7 ⎞ ⎠ ⎛
⎝⎛ y − 9 ⎞ ⎠ b ⎛ ⎝2xyx + 3 ⎠ ⎞⎛ ⎝⎛ 4xyx − 5 ⎞ ⎠.
⎛ ⎝3y −1 ⎞ ⎠ ⎛ ⎝2y −6 ⎞ ⎠.
3y − 1 by −6.
3y −1by2y.
3y −1
___________ ×2y −6
−18y +6
6y 2 − 2y ___________
6y 2 −20y +6
(5m − 7)(3m − 6).
(6b − 5)(7b − 3).

(b + 3)(2b 2 −5b +8) a b
a
b
(2b 2 −5b +8)
(2b 2 −5b +8) b
⎛
⎝y
−3 ⎞⎛ ⎛
⎠ ⎝y
2 − 5y + 2 ⎞ ⎠ a b
(x + 4)(2x 2 − 3x + 5) a b

(a + b) 2 = ___ + ___ + ___
(a + b) 2 = a 2 + ___ + ___
(a + b) 2 = ___ + ___ + b 2
(a + b) 2 = ___ + 2ab + ___
(a − b) 2 = ___ − 2ab + ___
a (x +5) 2 b ⎛ ⎝2x −3y ⎞ ⎠2 .
a

a (x + 9) 2 b (2c − d) 2 .
a ⎛
⎝y
+ 11 ⎞ ⎠ 2 b ⎛ ⎝4x − 5y ⎞2 ⎠ .
(a − b), (a + b).
(a − b), (a + b).
(a + b)(a − b) =a 2 − ___
(a + b)(a − b) =a 2 − b 2
a 2 − b 2 .

a (2x +5)(2x −5) b (5m −9n)(5m +9n).
a
2x.
5.
b
a (6x + 5)(6x − 5) b ⎛ ⎝ ⎛ 4p − 7q ⎞ ⎠ ⎞⎛ ⎝⎛ 4p + 7q ⎞ ⎠ ⎞ .
a (2x + 7)(2x − 7) b ⎛ ⎝ ⎛ 3x − y ⎞ ⎠ ⎞⎛ ⎝⎛ 3x + y ⎞ ⎠ ⎞ .

(a + b) 2 = a 2 +2ab + b 2 (a − b)(a + b) = a 2 − b 2
(a − b) 2 = a 2 −2ab + b 2
a (2x −3)(2x +3) b (8x −5) 2 c (6m +7) 2 d (5x −6)(6x +5).
a (2x −3)(2x +3)
b (8x −5) 2
c (6m +7) 2

d (5x −6)(6x +5)
(5x −6)(6x +5)
Use FOIL.
30x 2 +25x −36x −30
Simplify.
30x 2 −11x −30
a (9b − 2)(2b + 9) b ⎛ ⎝ ⎛ 9p − 4 ⎞ ⎠ ⎞ 2 c ⎛ ⎝ ⎛ 7y + 1 ⎞ ⎠ ⎞ 2 d (4r − 3)(4r + 3).
a (6x + 7) 2 b (3x − 4)(3x + 4) c (2x − 5)(5x − 2) d (6n − 1) 2 .
f (x) g(x),
⎛
⎝f · g ⎞ ⎠(x) = f (x) · g(x)
f (x) = x +2 g(x) = x 2 −3x −4, a ⎛ ⎝f · g ⎞ ⎠(x) b ⎛ ⎝f · g ⎞ ⎠(2).
a
⎛
⎝f · g ⎞ ⎠(x) = f (x) · g(x)
f (x)and g(x).
⎛
⎝f · g ⎞ ⎠(x) = (x +2) ⎛ ⎝x 2 −3x −4 ⎞ ⎠
⎛
⎝f · g ⎞ ⎠(x) = x ⎛ ⎝x 2 −3x −4 ⎞ ⎠ +2 ⎛ ⎝x 2 −3x −4 ⎞ ⎠
⎛
⎝f · g ⎞ ⎠(x) = x 3 −3x 2 −4x +2x 2 −6x −8
⎛
⎝f · g ⎞ ⎠(x) = x 3 − x 2 −10x −8
b a ⎛ ⎝f · g ⎞ ⎠(x) ⎛ ⎝f · g ⎞ ⎠(2).

⎛
⎝f · g ⎞ ⎠(x) = x 3 − x 2 −10x −8
⎛ ⎝f · g ⎞ ⎠(2), x =2.
⎛
⎝f · g ⎞ ⎠(2) =2 3 −2 2 −10·2−8
⎛
⎝f · g ⎞ ⎠(2) =8−4−20−8
⎛
⎝f · g ⎞ ⎠(2) = −24
f (x) = x − 5 g(x) = x 2 − 2x + 3, a ⎛
⎝f
· g ⎞ ⎠(x) b ⎛ ⎝f
· g ⎞ ⎠(2).
f (x) = x − 7 g(x) = x 2 + 8x + 4, a ⎛
⎝f
· g ⎞ ⎠(x) b ⎛ ⎝f
· g ⎞ ⎠(2).

a ⎛ ⎝6y 7⎞ ⎠ ⎛ ⎝−3y 4⎞ ⎠
b ⎛ ⎝ 4 7 rs2 ⎞⎛ ⎠⎝ 14rs 3⎞ ⎠
a ⎛ ⎝−10x 5⎞ ⎠ ⎛ ⎝−3x 3⎞ ⎠
b ⎛ ⎝ 5 8 x3 y ⎞ ⎠ ⎛ ⎝24x 5 y ⎞ ⎠
a (−8u 6 )(−9u)
b ⎛ ⎝ 2 3 x2 y ⎞ ⎠ ⎛ ⎝ 3 4 xy2 ⎞
⎠
a (−6c 4 )(−12c)
b ⎛ ⎝ 3 5 m3 n 2 ⎞⎛5 ⎠⎝ m 2 n 3 ⎞
9 ⎠
a −8x(x 2 +2x − 15)
b 5pq 3 (p 2 −2pq +6q 2 )
a −5t(t 2 +3t − 18);
b 9r 3 s(r 2 −3rs +5s 2 )
a −8y(y 2 +2y − 15)
b −4y 2 z 2 (3y 2 +12yz − z 2 )
a −5m(m 2 +3m − 18)
b −3x 2 y 2 (7x 2 +10xy − y 2 )
a b c
(w +5)(w +7) (y +9)(y +3) (4p + 11)(5p −4)
(7q + 4)(3q −8)
(x +8)(x +3) (y −6)(y −2) (2t − 9)(10t +1)
(6p +5)(p +1) (q −5)(q +8) (m + 11)(m −4)
(7m +1)(m −3) (3r − 8)(11r +1) (x 2 +3)(x +2)
(y 2 −4)(y +3) (5ab − 1)(2ab +3) (2xy + 3)(3xy +2)
(x 2 +8)(x 2 −5) (y 2 −7)(y 2 −4) (6pq − 3)(4pq −5)
(3rs −7)(3rs −4)

a b
(x +5)(x 2 +4x +3) (u +4)(u 2 +3u +2) (y + 8)(4y 2 + y −7)
(a + 10)(3a 2 + a −5) (y 2 −3y + 8)(4y 2 + y −7)
(2a 2 −5a + 10)(3a 2 + a −5)
(w −7)(w 2 −9w + 10) (p −4)(p 2 −6p +9) (3q +1)(q 2 −4q −5)
(6r +1)(r 2 −7r −9)
(w +4) 2 (q + 12) 2 ⎛ ⎝3x − y ⎞ 2
⎠
⎛ ⎝2y −3z ⎞ 2
⎠
⎛ ⎝ y + 1 2
⎞
4⎠
⎛ ⎝ x + 2 2
⎞
3⎠
⎛ ⎝ 1 5 x − 1 2
7 y⎞ ⎠
⎛ ⎝ 1 8 x − 1 2
9 y⎞ ⎠
(3x 2 +2) 2
(5u 2 +9) 2 (4y 3 −2) 2 (8p 3 −3) 2
(5k + 6)(5k −6) (8j + 4)(8j −4) (11k + 4)(11k −4)
(9c + 5)(9c −5) (9c −2d)(9c +2d) (7w +10x)(7w −10x)
⎛ ⎝ m + 2 3 n⎞ ⎠ ⎛ ⎝ m − 2 3 n⎞ ⎠
⎛ ⎝ p + 4 5 q⎞ ⎠ ⎛ ⎝ p − 4 5 q⎞ ⎠
(ab −4)(ab +4)
(xy −9)(xy +9)
(12p 3 −11q 2 )(12p 3 +11q 2 )
(15m 2 −8n 4 )(15m 2 +8n 4 )
⎛ ⎝p −3 ⎞ ⎠ ⎛ ⎝p +3 ⎞ ⎠ (t −9) 2 (m + n) 2
(2x + y)(x −2y) (2r + 12) 2 (3p + 8)(3p −8)
(7a + b)(a −7b) (k −6) 2 (a 5 −7b) 2
(x 2 +8y)(8x − y 2 ) (r 6 + s 6 )(r 6 − s 6 ) (y 4 +2z) 2
(x 5 + y 5 )(x 5 − y 5 ) (m 3 −8n) 2 (9p +8q) 2

(r 2 − s 3 )(r 3 + s 2 )
⎛ ⎝10y −6 ⎞ ⎠ + ⎛ ⎝4y −7 ⎞ ⎠
⎛ ⎝15p −4 ⎞ ⎠ + ⎛ ⎝3p −5 ⎞ ⎠
⎛ ⎝x 2 −4x −34 ⎞ ⎠ − ⎛ ⎝x 2 +7x −6 ⎞ ⎠
⎛ ⎝j 2 −8j −27 ⎞ ⎠ − ⎛ ⎝j 2 +2j −12 ⎞ ⎠
( 1 5 f 8 )(20 f 3 ) ( 1 4 d5 )(36d 2 )
(4a 3 b)(9a 2 b 6 ) (6m 4 n 3 )(7mn 5 )
−5m(m 2 +3m − 18) 5q 3 (q 2 −2q +6)
(s −7)(s +9)
⎛ ⎝y 2 −2y ⎞ ⎠ ⎛ ⎝y +1 ⎞ ⎠
(5x − y)(x −4)
(6k −1) ⎛ ⎝k 2 +2k −4 ⎞ ⎠
⎛ ⎝3x −11y ⎞ ⎠ ⎛ ⎝3x −11y ⎞ ⎠ (11 − b)(11 + b)
(rs − 2 7 )(rs + 2 7 ) (2x2 −3y 4 )(2x 2 +3y 4 )
(m −15) 2 (3d +1) 2
(4a +10) 2 ⎛ ⎝ 3z + 1 2
5⎞
⎠
f (x) = x +2
f (x) = x −1
g(x) =3x 2 −2x +4,
a ⎛ ⎝f · g ⎞ ⎠(x) b ⎛ ⎝f · g ⎞ ⎠(−1)
g(x) =4x 2 +3x −5,
a ⎛ ⎝f · g ⎞ ⎠(x) b ⎛ ⎝f · g ⎞ ⎠(−2)
f (x) =2x −7
g(x) =2x +7, a ⎛ ⎝f · g ⎞ ⎠(x) b ⎛ ⎝f · g ⎞ ⎠(−3)
f (x) =7x −8 g(x) =7x +8,
a ⎛ ⎝f · g ⎞ ⎠(x) b ⎛ ⎝f · g ⎞ ⎠(−2)
f (x) = x 2 −5x +2
g(x) = x 2 −3x −1, a ⎛ ⎝f · g ⎞ ⎠(x) b ⎛ ⎝f · g ⎞ ⎠(−1)
f (x) = x 2 +4x −3
g(x) = x 2 +2x +4, a ⎛ ⎝f · g ⎞ ⎠(x) b ⎛ ⎝f · g ⎞ ⎠(1)
(x +2)(x −2)
(y +7)(y −7)
(w +5)(w −5)

(p +3)(p +3)
(q +6)(q +6)
(r +1)(r +1)
(a + b) 2
(a − b)(a + b)
a
b

3
d
+ x d .
30xyx
3
.
5xyx
8a 2 + 12a + 1 + 3a 2 − 5a + 4.
54a 2 b 3 ÷ ⎛ ⎝−6ab 5⎞ ⎠.

54a 2 b 3 ÷ ⎛ ⎝−6ab 5⎞ ⎠
54a 2 b 3
−6ab 5
54
−6 · a2 a · b3
b 5
−9 · a · 1
b 2
− 9a
b 2
−72a 7 b 3 ÷ ⎛ ⎝ ⎛ 8a 12 b 4⎞ ⎠ ⎞ .
−63c 8 d 3 ÷ ⎛ ⎝ ⎛ 7c 12 d 2⎞ ⎠ ⎞ .
14x 7 y 12
21x 11 y 6.
14
21
14x 7 y 12
21x 11 y 6
2y 6
3x 4
28x 5 y 14
49x 9 y 12.
30m 5 n 11
48m 10 n 14.

y 5 + 2 5
y +2
5 .
y +2
5
y 5 + 2 5 .
c ≠0, a + b c = a c + b c .
⎛ ⎝18x 3 y −36xy 2⎞ ⎠ ÷ ⎛ ⎝−3xy ⎞ ⎠.
⎛
⎝18x 3 y −36xy 2⎞ ⎠ ÷ ⎛ ⎝−3xy ⎞ ⎠
18x 3 y −36xy 2
−3xy
18x 3 y
−3xy − 36xy2
−3xy
−6x 2 +12y
⎛ ⎝ ⎛ 32a 2 b − 16ab 2⎞ ⎠ ⎞ ÷ (−8ab).
⎛ ⎝ ⎛ −48a 8 b 4 − 36a 6 b 5⎞ ⎠ ⎞ ÷ ⎛ ⎝ ⎛ −6a 3 b 3⎞ ⎠ ⎞ .
35·25
875 ✓

⎛ ⎝x 2 +9x +20 ⎞ ⎠ ÷ (x +5).
x 2 x.
x x 2
x, x
x x +5.
x 2 +5x x 2 +9x.
4x x.
x 4x
4
x +5.
4x +20 4x + 20.
(x +4)(x +5)
x 2 +9x +20✓
⎛
⎝y
2 +10y + 21 ⎞ ⎠ ÷ ⎛
⎝y
+ 3 ⎞ ⎠.
⎛ ⎝ ⎛ m 2 + 9m + 20 ⎞ ⎠ ⎞ ÷ (m + 4).
x 4 − x 2 +5x −6. x 3 0x 3
⎛ ⎝x 4 − x 2 +5x −6 ⎞ ⎠ ÷ (x +2).

x 3 0x 3
x 4 x.
x 3 , x 3
x 3 x +2.
−2x 3 x.
−2x 2 ,
x 2
−2x 2 x +1.
3x 2 x.
3x, x
3x x +1.
−x x.
−1,
−1 x +1.
(x +2) ⎛ ⎝ x3 −2x 2 +3x −1−
x +2
4 ⎞
⎠
x 4 − x 2 +5x −6.
⎛
⎝x
4 − 7x 2 + 7x + 6 ⎞ ⎠ ÷ (x + 3).
⎛
⎝x
4 − 11x 2 − 7x − 6 ⎞ ⎠ ÷ (x + 3).
2a +3.

⎛ ⎝8a 3 +27 ⎞ ⎠ ÷ (2a +3).
(2a +3) ⎛ ⎝4a 2 −6a +9 ⎞ ⎠.
8a 3 + 27.
⎛
⎝x
3 − 64 ⎞ ⎠ ÷ (x − 4).
⎛ ⎝ ⎛ 125x 3 − 8 ⎞ ⎠ ⎞ ÷ (5x − 2).
x x 2
−x 2 −4x
−5
Synthetic division only works when the divisor is of the form x − c.
2x 3 +3x 2 + x +8 x +2.

x.
x − c
2x 2 −1x +3
2 −1 3
2x 2 −1x +3.
(quotient)(divisor) + remainder = dividend
⎛
⎝2x 2 −1x +3 ⎞ ⎠(x +2) +2 = ? 2x 3 +3x 2 + x +8
2x 3 − x 2 +3x +4x 2 −2x +6+2 = ? 2x 3 +3x 2 + x +8
2x 3 +3x 2 + x +8 = 2x 3 +3x 2 + x +8✓
3x 3 + 10x 2 + 6x − 2 x + 2.

4x 3 + 5x 2 − 5x + 3 x + 2.
x 4 −16x 2 +3x +12 x +4.
x 4 −16x 2 +3x +12 x 3
x 3 x − c x − (−4).
1 −4 0 3, x 3 −4x 2 +3.
x 4 − 16x 2 + 5x + 20 x + 4.
x 4 − 9x 2 + 2x + 6 x + 3.
f (x) g(x), g(x) ≠0,
⎛
⎝g f ⎞ f (x)
⎠
(x) =
g(x)
f (x) = x 2 −5x −14 g(x) = x +2, a ⎛ ⎝ f g ⎞ ⎠ (x) b ⎛ ⎝ f g ⎞ ⎠ (−4).
a

f (x) g(x).
⎛
⎝ f g ⎞ ⎠ (x) = x2 −5x −14
x +2
⎛
⎝g f ⎞ ⎠
(x) = x −7
b a ⎛ ⎝ f g ⎞ ⎠ (x) ⎛ ⎝ f g ⎞ ⎠ (−4).
⎛
⎝g f ⎞ ⎠
(x) = x −7
⎛ ⎝ f g ⎞ ⎠ (−4), x =−4. ⎛
⎝
f
g
⎞
⎠
(−4) =−4−7
⎛
⎝g f ⎞ ⎠
(−4) = −11
f (x) = x 2 − 5x − 24 g(x) = x + 3, a ⎛
⎝
f ⎞ g⎠
(x) b ⎛ f ⎞ ⎝ g⎠
(−3).
f (x) = x 2 − 5x − 36 g(x) = x + 4, a ⎛
⎝
f ⎞ g⎠
(x) b ⎛ f ⎞ ⎝ g⎠
(−5).
x − c, c, f (c),
x − c fc
x 4 − x 2 +5x −6 x − (−2) −4 f (x) = x 4 − x 2 +5x −6 −4
3x 3 −2x 2 −10x +8 x −2 f (x) =3x 3 −2x 2 −10x +8
x 4 −16x 2 +3x +15 x − (−4) f (x) = x 4 −16x 2 +3x +15
f (x), q(x)
x − c,

c,
f (x) x − c, f (c).
f (x) = x 3 +3x +19 x +2.
x − c x +2 x − (−2).
c −2.
f (c) f (−2).
f (−2),
x =−2.
f (x) =x 3 +3x +19
x +2.
f (x) = x 3 + 4x + 15 x + 2.

f (x) = x 3 − 7x + 12 x + 3.
8a 3 +27 2a +3 4a 2 −6a +9.
4a 2 −6a +9 2a +3 8a 3 +27
⎛
⎝4a 2 −6a +9 ⎞ ⎠(2a +3) =8a 3 +27
4a 2 −6a +9 2a +3 8a 3 + 27.
x − c, f (x), x − c
f (x).
x − c
f (x) x − c
f (x),
x − c f (x), f (c) =0
f (c) =0, x − c f (x)
x −4 f (x) = x 3 − 64.
x −4 f (x) = x 3 −64 f (4) =0.
f (x) = x 3 −64
f (4) x =4.
f (4) =4 3 −64
f (4) =64−64
f (4) =0
f (4) =0, x −4 f (x) = x 3 − 64.
x − 5 f (x) = x 3 − 125.
x − 6 f (x) = x 3 − 216.

15r 4 s 9 ÷ ⎛ ⎝15r 4 s 9⎞ ⎠ 20m 8 n 4 ÷ ⎛ ⎝30m 5 n 9⎞ ⎠ 18a 4 b 8
−27a 9 b 5
45x 5 y 9
−60x 8 y 6
⎛
⎝10m 5 n 4⎞ ⎠ ⎛ ⎝5m 3 n 6⎞ ⎠
25m 7 n 5
⎛
⎝−18p 4 q 7⎞ ⎠ ⎛ ⎝−6p 3 q 8⎞ ⎠
−36p 12 q 10
⎛
⎝6a 4 b 3⎞ ⎠ ⎛ ⎝4ab 5⎞ ⎠
⎛
⎝12a 2 b ⎞ ⎠ ⎛ ⎝a 3 b ⎞ ⎠
⎛
⎝4u 2 v 5⎞ ⎠ ⎛ ⎝15u 3 v ⎞ ⎠
⎛
⎝12u 3 v ⎞ ⎠ ⎛ ⎝u 4 v ⎞ ⎠
⎛ ⎝9n 4 +6n 3⎞ ⎠ ÷3n ⎛ ⎝8x 3 +6x 2⎞ ⎠ ÷2x ⎛ ⎝63m 4 −42m 3⎞ ⎠ ÷ ⎛ ⎝−7m 2⎞ ⎠
⎛ ⎝48y 4 −24y 3⎞ ⎠ ÷ ⎛ ⎝−8y 2⎞ ⎠
66x 3 y 2 − 110x 2 y 3 −44x 4 y 3
11x 2 y 2
72r 5 s 2 + 132r 4 s 3 −96r 3 s 5
12r 2 s 2
10x 2 +5x −4
−5x
20y 2 +12y −1
−4y
⎛ ⎝y 2 +7y +12 ⎞ ⎠ ÷ ⎛ ⎝y +3 ⎞ ⎠ ⎛ ⎝a 2 −2a −35 ⎞ ⎠ ÷ (a +5)
⎛
⎝6m 2 −19m −20 ⎞ ⎠ ÷ (m −4)
⎛ ⎝4x 2 −17x −15 ⎞ ⎠ ÷ (x −5) ⎛ ⎝q 2 +2q +20 ⎞ ⎠ ÷ ⎛ ⎝q +6 ⎞ ⎠ ⎛ ⎝p 2 +11p +16 ⎞ ⎠ ÷ ⎛ ⎝p +8 ⎞ ⎠
⎛ ⎝3b 3 + b 2 +4 ⎞ ⎠ ÷ (b +1) ⎛ ⎝2n 3 −10n +28 ⎞ ⎠ ÷ (n +3) ⎛ ⎝z 3 +1 ⎞ ⎠ ÷ (z +1)
⎛ ⎝m 3 + 1000 ⎞ ⎠ ÷ (m +10) ⎛ ⎝64x 3 −27 ⎞ ⎠ ÷ (4x −3) ⎛ ⎝125y 3 −64 ⎞ ⎠ ÷ ⎛ ⎝5y −4 ⎞ ⎠
x 3 −6x 2 +5x +14 x +1 x 3 −3x 2 −4x +12 x +2
2x 3 −11x 2 +11x +12 x −3
2x 3 −11x 2 +16x −12 x −4
x 4 −5x 2 +13x +3 x +3 x 4 + x 2 +6x −10 x +2

2x 4 −9x 3 +5x 2 −3x −6 x −4
3x 4 −11x 3 +2x 2 +10x +6 x −3
f (x) = x 2 −13x +36 f (x) = x 2 −15x +54
g(x) = x −4, a ⎛ ⎝g f ⎞ ⎠ (x) b ⎛ ⎝g f ⎞ ⎠ (−1) g(x) = x −9, a ⎛ ⎝g f ⎞ ⎠ (x) b ⎛ ⎝g f ⎞ ⎠ (−5)
f (x) = x 3 + x 2 −7x +2 f (x) = x 3 +2x 2 −19x +12
g(x) = x −2, a ⎛ ⎝g f ⎞ ⎠ (x) b ⎛ ⎝g f ⎞ ⎠ (2) g(x) = x −3, a ⎛ ⎝g f ⎞ ⎠ (x) b ⎛ ⎝g f ⎞ ⎠ (0)
f (x) = x 2 −5x +2
g(x) = x 2 −3x −1, a ⎛ ⎝f · g ⎞ ⎠(x) b ⎛ ⎝f · g ⎞ ⎠(−1)
f (x) = x 2 +4x −3
g(x) = x 2 +2x +4, a ⎛ ⎝f · g ⎞ ⎠(x) b ⎛ ⎝f · g ⎞ ⎠(1)
f (x) = x 3 −8x +7 x +3 f (x) = x 3 −4x −9 x +2
f (x) =2x 3 −6x −24 x −3
f (x) =7x 2 −5x −8 x −1
x−c
x +3
x 3 +8x 2 +21x +18
x −2
x 3 −7x 2 +7x −6
x +4
x 3 + x 2 −14x +8
x −3
x 3 −7x 2 +11x +3
48y +6
48y +6
6
=48y.
10x2 + x −12
2x
x 2 +5x +6 x −2.

a
b

(a − b), (a + b).
ax m ,
ax m ,

m th
a m
a m · a n = a m + n
a a ≠0,
a 0 =1.
a m
a n = am − n , m > n and a m
a n = 1
a n − m, n > m
a ≠0, a −n = 1 a n 1
a −n = an .
a, b a ≠0,b ≠0 n
a m, n
⎛
⎝ a −n
⎞
b⎠
= ⎛b ⎝a ⎞ n
⎠
(a m ) n = a m · n
(ab) m = a m b m
a b ≠0, m
⎛
⎝ a m
⎞
b⎠
= am
b m

a m · a n = a m + n
(a m ) n = a m · n
(ab) n = a m b m
a m
a n = am − n , a ≠0
a 0 =1,a ≠0
m
⎛a⎞ ⎝b⎠ = am
b
m, b ≠0
a −n = 1 a n 1 −n = an
a
−n
⎛a⎞ ⎝b⎠ = ⎛b ⎝a ⎞ n
⎠
a ×10 n where 1 ≤ a < 10 and n is an integer.
n,
10 n .
10 −n .
n,
n
n
|n|

a, b
(a + b) 2 = a 2 +2ab + b 2 (a − b) 2 = a 2 −2ab + b 2
(a − b)(a + b) = a 2 − b 2
f (x) g(x),
⎛
⎝f · g ⎞ ⎠(x) = f (x) · g(x)
f (x) g(x), g(x) ≠0,
⎛
⎝g f ⎞ f (x)
⎠
(x) =
g(x)

f (x) x − c, f (c).
f (x),
x − c f (x), f (c) =0
f (c) =0, x − c f (x)
16x 2 −40x −25 5m +9
−15 y 2 +6y 3 +9y 4
4p +11p −8y 3 −5y 3
⎛
⎝4a 2 +9a −11 ⎞ ⎠ + ⎛ ⎝6a 2 −5a +10 ⎞ ⎠
⎛
⎝8m 2 +12m −5 ⎞ ⎠ − ⎛ ⎝2m 2 −7m −1 ⎞ ⎠
⎛
⎝y 2 −3y +12 ⎞ ⎠ + ⎛ ⎝5y 2 −9 ⎞ ⎠
⎛
⎝5u 2 +8u ⎞ ⎠ − (4u −7)
8q 3 −27 q 2 +6q −2. x 2 +6x +8
x 2 −8x + 15.
17mn 2 −(−9mn 2 )+3mn 2 18a −7b −21a
2pq 2 −5p −3q 2 (6a 2 +7)+(2a 2 −5a −9)
(3p 2 −4p −9)+(5p 2 + 14) (7m 2 −2m −5)−(4m 2 + m −8)
(7b 2 −4b +3)−(8b 2 −5b −7) ⎛ ⎝8y 2 − y +9 ⎞ ⎠ ⎛ ⎝11y 2 −9y −5 ⎞ ⎠
⎛
⎝z 2 −4z −12 ⎞ ⎠
⎛
⎝x 3 − x 2 y ⎞ ⎠ − ⎛ ⎝4xy 2 − y 3⎞ ⎠ + ⎛ ⎝3x 2 y − xy 2⎞ ⎠
⎛
⎝3z 2 +2z −11 ⎞ ⎠
⎛
⎝x 3 −2x 2 y ⎞ ⎠ − ⎛ ⎝xy 2 −3y 3⎞ ⎠ − ⎛ ⎝x 2 y −4xy 2⎞ ⎠

f (x) =7x 2 −3x +5
a f (5) b f (−2) c f (0)
g(x) =15−16x 2 ,
a g(−1) b g(0) c g(2)
h(t) = −16t 2 + 640
t =6.
p
R(p) =−5p 2 + 360p.
p =10
a b c d
f (x) =2x 2 −4x −7 g(x) =2x 2 − x +5 f (x) =4x 3 −3x 2 + x −1 g(x) =8x 3 −1
p 3 · p 10 2·2 6 a · a 2 · a 3
x · x 8 y a · y b 2 8
2 2
a 6
a n 3
n 12 1
x 5
3 0 y 0 (14t) 0
12a 0 −15b 0
6 −2 (−10) −3 5·2 −4
(8n) −1 y −5 10 −3
1
a −4 1
6 −2 −5 −3
−3
⎛
⎝ −1 ⎞
5⎠
− ⎛ ⎝ 1 −3
⎞
2⎠
(−5) −3
⎛
⎝ 5 −2
⎞
9⎠
⎛
⎝ −3 x ⎞ −3
⎠

⎛
⎝y 4⎞ 3
⎠
⎛
⎝3 2⎞ 5
⎠
⎛
⎝a 10⎞ y
⎠
x −3 · x 9 r −5 · r −4
⎛
⎝uv −3⎞ ⎠ ⎛ ⎝u −4 v −2⎞ ⎠
⎛
⎝m 5⎞ −1
⎠
p 5 · p −2 · p −4
(k −2 ) −3 q 4
q 20 b 8
b −2
n −3
n −5
(−5ab) 3
⎛
⎝−4pq ⎞ 0
⎠
⎛
⎝−6x 3⎞ −2
⎠
⎛
⎝3y −4⎞ 2
⎠
⎛
⎝5x
3 −2
⎞
⎛
⎠
⎜ 3xy2 4
2
⎞
⎛
⎟
⎝
z
⎜ 4p−3 ⎞
⎠
⎝ q 2 ⎟
⎠
(x 2 y) 2 (3xy 5 ) 3
⎛
⎝−3a −2⎞ ⎛⎝
⎠4
2a 4⎞ 2
⎠
⎛
⎝−6a 2⎞ 3
⎠
2 −1
⎛
⎜ 3xy3 ⎞ ⎛
⎝4x 4 y −2 ⎟
⎠ ⎝ ⎜ 6xy4 ⎞
8x 3 y −2 ⎟
⎠
2.568 0.00814
2.9 × 10 4 3.75 × 10 −1 9.413 × 10 −5
⎛
⎝3×10 7⎞ ⎠ ⎛ ⎝2×10 −4⎞ ⎠
⎛
⎝1.5 × 10 −3⎞ ⎠ ⎛ ⎝4.8 × 10 −1⎞ ⎠ 6×10 9
2×10 −1
9×10 −3
1×10 −6

⎛
⎝−6p 4⎞ ⎠ ⎛ ⎝9p ⎞ ⎠
⎛
⎝ 1 3 c2 ⎞
⎠
⎛
⎝ 30c 8⎞ ⎠
⎛
⎝8x 2 y 5⎞ ⎠ ⎛ ⎝7xy 6⎞ ⎠
⎛
⎝ 2 3 m3 n 6⎞ ⎠ ⎛ ⎝ 1 6 m4 n 4 ⎞
⎠
7(10 − x) a 2 ⎛ ⎝a 2 −9a −36 ⎞ ⎠ −5y ⎛ ⎝125y 3 −1 ⎞ ⎠
(4n −5) ⎛ ⎝2n 3⎞ ⎠
a b c
(a +5)(a +2)
⎛
⎝y −4 ⎞ ⎠ ⎛ ⎝y +12 ⎞ ⎠ (3x +1)(2x −7)
⎛
⎝6p −11 ⎞ ⎠ ⎛ ⎝3p −10 ⎞ ⎠
(n +8)(n +1) (k +6)(k −9) (5u −3)(u +8)
⎛
⎝2y −9 ⎞ ⎠ ⎛ ⎝5y −7 ⎞ ⎠
⎛
⎝p +4 ⎞ ⎠ ⎛ ⎝p +7 ⎞ ⎠ (x −8)(x +9)
(3c +1)(9c −4) (10a −1)(3a −3)
a b
(x +1) ⎛ ⎝x 2 −3x −21 ⎞ ⎠ (5b −2) ⎛ ⎝3b 2 + b −9 ⎞ ⎠
(m +6) ⎛ ⎝m 2 −7m −30 ⎞ ⎠
⎛
⎛
⎝4y −1 ⎞ ⎠⎝6y 2 −12y +5 ⎞ ⎠
⎛
⎝2x − y ⎞ 2
⎠
⎛
⎝ x + 3 2
⎞
(8p 3 −3) 2
4⎠
(5p +7q) 2

(3y + 5)(3y −5) (6x + y)(6x − y)
⎛
⎝ a + 2 3 b⎞ ⎠ ⎛ ⎝ a − 2 3 b⎞ ⎠
(12x 3 −7y 2 )(12x 3 +7y 2 ) (13a 2 −8b 4 )(13a 2 +8b 4 )
72p 12 ÷8p 3 −26a 8 ÷ ⎛ ⎝2a 2⎞ ⎠
45y 6
−15y 10
−30x 8
−36x 9 28a 9 b
7a 4 b 3 11u 6 v 3
55u 2 v 8
⎛
⎝5m 9 n 3⎞ ⎠ ⎛ ⎝8m 3 n 2⎞ ⎠
⎛
⎝10mn 4⎞ ⎠ ⎛ ⎝m 2 n 5⎞ ⎠
(42r 2 s 4 )(54rs 2 )
(6rs 3 )(9s)
⎛
⎝54y 4 −24y 3⎞ ⎠ ÷ ⎛ ⎝−6y 2⎞ ⎠
63x 3 y 2 −99x 2 y 3 −45x 4 y 3 12x 2 +4x −3
−4x
9x 2 y 2
⎛
⎝4x 2 −21x −18 ⎞ ⎠ ÷ (x −6)
⎛
⎝y 2 +2y +18 ⎞ ⎠ ÷ ⎛ ⎝y +5 ⎞ ⎠
⎛
⎝n 3 −2n 2 −6n +27 ⎞ ⎠ ÷ (n +3)
⎛
⎝a 3 −1 ⎞ ⎠ ÷ (a +1)
x 3 −3x 2 −4x +12
x +2
2x 3 −11x 2 +11x +12
x −3
x 4 + x 2 +6x −10
x +2
f (x) = x 2 −15x +54 f (x) = x 3 + x 2 −7x +2
g(x) = x −9, a ⎛ ⎝g f ⎞ ⎠ (x)
g(x) = x −2, a ⎛ ⎝g f ⎞ ⎠ (x)
b ⎛ ⎝g f ⎞ ⎠ (−2) b ⎛ ⎝g f ⎞ ⎠ (3)

f (x) = x 3 −4x −9 x +2 f (x) =2x 3 −6x −24 x −3
x−c
x −2
x 3 −7x 2 +7x −6
x −3
x 3 −7x 2 +11x +3

8y 4 −3y 2 +1
⎛
⎝5a 2 +2a −12 ⎞ ⎠ ⎛ ⎝9a 2 +8a −4 ⎞ ⎠
⎛
⎝10x 2 −3x +5 ⎞ ⎠ − ⎛ ⎝4x 2 −6 ⎞ ⎠
a
b
⎛
⎝ −3 3
⎞
4⎠
x −3 x 4 5 6
5 8
⎛
⎝47a 18 b 23 c 5⎞ 0
⎠
4 −1
⎛
⎝2y ⎞ −3
⎠
p −3 · p −8 x 4
x −5
⎛
⎝3x −3⎞ 2
⎠
24r 3 s
6r 2 s 7
⎛
⎜ x4 y 9 2
⎞
⎝ x −3 ⎟
⎠
⎛
⎝8xy 3⎞ ⎠ ⎛ ⎝−6x 4 y 6⎞ ⎠
4u ⎛ ⎝u 2 −9u +1 ⎞ ⎠
(m +3)(7m −2) (n −8) ⎛ ⎝n 2 −4n +11 ⎞ ⎠
(4x −3) 2
⎛
⎝5x +2y ⎞ ⎠ ⎛ ⎝5x −2y ⎞ ⎠
⎛
⎝15xy 3 −35x 2 y ⎞ ⎠ ÷5xy
⎛
⎝3x 3 −10x 2 +7x +10 ⎞ ⎠ ÷ (3x +2)
x +3
x 3 +8x 2 +21x + 18.
a
b
5.25 × 10 −4
⎛
⎝2.4 × 10 8⎞⎛ ⎠⎝2×10 −5⎞ ⎠ 9×10 4
3×10 −1 f (x) =6x 2 −3x −9
a f (3) b f (−2) c f (0)
f (x) =2x 2 −3x −5
g(x) =3x 2 −4x +1,
a ⎛ ⎝f + g ⎞ ⎠(x) b ⎛ ⎝f + g ⎞ ⎠(1)
c ⎛ ⎝f − g ⎞ ⎠(x) d ⎛ ⎝f − g ⎞ ⎠(−2)
f (x) =3x 2 −23x −36
g(x) = x −9,
h(t) = −16t 2 + 240
a ⎛ ⎝g f ⎞ ⎠ (x) b ⎛ ⎝g f ⎞ t
⎠ (3)
t =3.

−3a(7a + 8b).
21x 3 ,9x 2 ,15x.
21x 3 9x 2 15x 3x

25m 4 , 35m 3 , 20m 2 .
14x 3 , 70x 2 , 105x.
2·6 3·4.
3x 2 +15x,
3x(x +5).
a(b + c) = ab + ac and ab + ac = a(b + c)
8m 3 −12m 2 n +20mn 2 .
9xyx 2 + 6x 2 y 2 + 21y 3 .
3p 3 − 6p 2 q + 9pq 3 .

Noun: 7 is a factor of 14
Verb: factor 3 from 3a +3
5x 3 −25x 2 .
5x 3 25x 2 .
5x 2 (x −5)
5x 2 · x −5x 2 ·5
5x 3 −25x 2 ✓
2x 3 + 12x 2 .
6y 3 − 15y 2 .
8x 3 y −10x 2 y 2 +12xy 3 .

8x 3 y, −10x 2 y 2 , and 12xy 3
2xy.
2xy.
2xy ⎛ ⎝4x 2 −5xy +6y 2⎞ ⎠
2xy ·4x 2 −2xy ·5xy +2xy ·6y 2
8x 3 y −10x 2 y 2 +12xy 3 ✓
15x 3 y − 3x 2 y 2 + 6xyx 3 .
8a 3 b + 2a 2 b 2 − 6ab 3 .
−4a 3 +36a 2 −8a.
−4a.
−4a ⎛ ⎝a 2 −9a +2 ⎞ ⎠
−4a · a 2 − (−4a) ·9a + (−4a) ·2
−4a 3 +36a 2 −8a ✓
−4b 3 + 16b 2 − 8b.
−7a 3 + 21a 2 − 14a.

3y ⎛ ⎝y +7 ⎞ ⎠ −4 ⎛ ⎝y +7 ⎞ ⎠.
y +7.
⎛ ⎝y +7 ⎞ ⎠.
4m(m + 3) − 7(m + 3).
8n(n − 4) + 5(n − 4).
xy +3y +2x +6.
xy
+ 8y + 3x + 24.
ab + 7b + 8a + 56.

a x 2 +3x −2x −6 b 6x 2 −3x −4x +2.
a
x 2 +3x −2x −6
x 2 +3x −2x −6
x(x +3) −2(x +3)
b
(x +3)(x −2)
6x 2 −3x −4x +2
6x 2 −3x −4x +2
3x(2x −1) −2(2x −1)
(2x −1)(3x −2)
a x 2 + 2x − 5x −10 b 20x 2 − 16x − 15x + 12.
a y 2 + 4y − 7y − 28 b 42m 2 − 18m − 35m + 15.

10p 3 q,12pq 2 8a 2 b 3 ,10ab 2 12m 2 n 3 ,30m 5 n 3
28x 2 y 4 ,42x 4 y 4 10a 3 ,12a 2 ,14a 20y 3 ,28y 2 ,40y
35x 3 y 2 ,10x 4 y,5x 5 y 3 27p 2 q 3 ,45p 3 q 4 ,9p 4 q 3
6m +9 14p +35 9n −63
45b −18 3x 2 +6x −9 4y 2 +8y −4
8p 2 +4p +2 10q 2 +14q +20 8y 3 +16y 2
12x 3 −10x 5x 3 −15x 2 +20x 8m 2 −40m +16
24x 3 −12x 2 +15x 24y 3 −18y 2 −30y 12xy 2 +18x 2 y 2 −30y 3
21pq 2 +35p 2 q 2 −28q 3 20x 3 y −4x 2 y 2 +12xy 3 24a 3 b +6a 2 b 2 −18ab 3
−2x −4 −3b +12 −2x 3 +18x 2 −8x
−5y 3 +35y 2 −15y −4p 3 q −12p 2 q 2 +16pq 2 −6a 3 b −12a 2 b 2 +18ab 2
5x(x +1) +3(x +1) 2x(x −1) +9(x −1) 3b(b −2) −13(b −2)
6m(m −5) −7(m −5)
ab +5a +3b +15 cd +6c +4d +24 8y 2 + y +40y +5
6y 2 +7y +24y +28 uv −9u +2v −18 pq −10p +8q −80
u 2 − u +6u −6 x 2 − x +4x −4 9p 2 +12p −15p −20
16q 2 +20q −28q −35 mn −6m −4n +24 r 2 −3r − r +3
2x 2 −14x −5x +35 4x 2 −36x −3x +27

−18xy 2 −27x 2 y −4x 3 y 5 − x 2 y 3 +12xy 4 3x 3 −7x 2 +6x −14
x 3 + x 2 + x +1 x 2 + xy +5x +5y 5x 3 −3x 2 +5x −3
y 4 , y 5 , y 10 ?
y a , y b , y c .
a
b

x 2 + bx + c
ax 2 + bx + c
ax 2 + bx + c
(3y + 4)(2y + 5).
−9(6); −9(−6).
x 2 + bx + c
x 2 + bx + c, x 2 +5x +6
(x +2)(x +3), (x + m) (x + n).

x 2 + bx + c.
x 2 + bx + c, (x + m) (x + n)
x 2 + bx + c
x 2 +11x + 24.
q 2 +10q + 24.
t 2 + 14t + 24.

x 2 + bx + c.
c, m · n = c
x 2 + bx + c
(x )(x )
b, m + n = b
(x + m)(x + n)
y 2 −11y + 28.
−11y,
−11.
y 2 −11y +28
y. (y )(y )
28
−1, −28
−2, −14
−4, −7
−1 + (−28) = −29
−2 + (−14) = −16
−4 + (−7) = −11*
−4, −7
⎛
⎝y −4 ⎞ ⎠ ⎛ ⎝y −7 ⎞ ⎠
⎛
⎝y −4 ⎞ ⎠ ⎛ ⎝y −7 ⎞ ⎠
y 2 −7y −4y +28
y 2 −11y +28✓

u 2 − 9u + 18.
y 2 − 16y + 63.
2x + x 2 − 48.
2x + x 2 −48
x 2 +2x −48
x. (x )(x )
−48
−1, 48
−2, 24
−3, 16
−4, 12
−6, 8
−1+48=47
−2+24=22
−3+16=13
−4+12=8
−6+8=2*
Use −6, 8 as the last terms of the binomials. (x −6)(x +8)
(x −6)(x +8)
x 2 −6q +8q −48
x 2 +2x −48✓
9m + m 2 + 18.
−7n + 12 + n 2 .
x 2 + bxy + cy 2 x 2 +12xy +36y 2 .
x 2 , x · x. y 2

2 −8rs −9s 2 .
r 2 −8rs −9s 2
r, s. (r s)(r s)
−9
1, −9 −1 + 9 = 8
−1, 9 1+(−9) =−8*
3, −3 3+(−3) =0
Use 1, −9 as coefficients of the last terms.
(r + s)(r −9s)
(r −9s)(r + s)
r 2 + rs −9rs −9s 2
r 2 −8rs −9s 2 ✓
a 2 − 11ab + 10b 2 .
m 2 − 13mn + 12n 2 .
u 2 −9uv −12v 2 .

u 2 −9uv −12v 2
u, v. (u v)(u v)
−12
1, −12
−1, 12
2, −6
−2, 6
3, −4
−3, 4
1+(−12) = −11
−1+12=11
2+(−6) =−4
−2+6=4
3+(−4) =−1
−3+4=1
−9
x 2 − 7xyx − 10y 2 .
p 2 + 15pq + 20q 2 .
x 2 + bx + c.
x 2 + bx + c
x 2 + bx + c
(x + m)(x + n)
When c is positive, m and n have the same sign.
b positive
b negative
m, n positive m, n negative
x 2 +5x +6 x 2 −6x +8
(x +2)(x +3) (x −4)(x −2)
same signs
same signs
When c is negative, m and n have opposite signs.
x 2 + x −12
x 2 −2x −15
(x +4)(x −3) (x −5)(x +3)
opposite signs
opposite signs
ax 2 + bx + c.

4x 3 +16x 2 −20x.
GCF = 4x.
4x 3 +16x 2 −20x
4x ⎛ ⎝x 2 +4x −5 ⎞ ⎠
4x(x )(x )
4x(x −1)(x +5)
−5
−1, 5
1, −5
−1 + 5 = 4*
1+(−5) =−4
4x(x −1)(x +5)
4x(x 2 +5x − x −5)
4x(x 2 +4x −5)
4x 3 +16x 2 −20x ✓
5x 3 + 15x 2 − 20x.
6y 3 + 18y 2 − 60y.
3x 2 +5x +2.
3x 2 +5x +2
( )( )
3x 2 . 3x 2 1x, 3x.

1x ·3x =3x 2 ?
5x,
(x +1)(3x +2)
3x 2 +2x +3x +2
3x 2 +5x +2✓
3x 2 +5x +2
(x +1)(3x +2)
3y 2 +22y +7.

2a 2 + 5a + 3.
4b 2 + 5b + 1.
ax 2 + bx + c
6b 2 −13b +5.

6b 2 −13b +5
(b −1)(6b −5) 6b 2 −11b +5
(b −5)(6b −1) 6b 2 −31b +5
(2b −1)(3b −5) 6b 2 −13b +5*
(2b −5)(3b −1) 6b 2 −17b +5
(2b −1)(3b −5)
(2b −1)(3b −5)
6b 2 −10b −3b +5
6b 2 −13b +5✓
8x 2 − 14x + 3.
10y 2 − 37y + 7.
18x 2 −37xy +15y 2 .

⎛
⎝2x −3y)(9x −5y ⎞ ⎠
⎛
⎝2x −3y)(9x −5y ⎞ ⎠
18x 2 −10xy −27xy +15y 2
18x 2 −37xy +15y 2 ✓
18x 2 − 3xyx − 10y 2 .
30x 2 − 53xyx − 21y 2 .
−10y 4 −55y 3 −60y 2 .

−5y 2 .
−5y 2 ⎛ ⎝y +4 ⎞ ⎠ ⎛ ⎝2y +3 ⎞ ⎠
−5y 2 ⎛ ⎝y +4 ⎞ ⎠ ⎛ ⎝2y +3 ⎞ ⎠
−5y 2 ⎛ ⎝2y 2 +8y +3y +12 ⎞ ⎠
−10y 4 −55y 3 −60y 2 ✓
15n 3 − 85n 2 +100n.
56q 3 + 320q 2 − 96q.
ax 2 + bx + c
ax 2 + bx + c
6x 2 +7x +2.

6x 2 + 13x + 2.
4y 2 + 8y + 3.
ax 2 + bx + c
Multiply to ac m · n = a · c
Add to b m + n = b
ax 2 + bx + c
ax 2 + mx + nx + c
10y 2 −55y + 70.

ac. ac =28
ac (−4)(−7) = 28
−4 + (−7) = −11
5 ⎛ ⎝y −2 ⎞ ⎠ ⎛ ⎝2y −7 ⎞ ⎠
5 ⎛ ⎝2y 2 −7y −4y +14 ⎞ ⎠
5 ⎛ ⎝2y 2 −11y +14 ⎞ ⎠
10y 2 −55y +70✓
16x 2 − 32x + 12.
18w 2 − 39w + 18.
ax 2 + bx + c
ax 2 + bx + c
ax 2 + bx + c, x 2 ,
x 4 −4x 2 −5.
x 2 x 4 ,
⎛
⎝x 2⎞ ⎠2
= x 4 ). u = x 2 , ax 2 + bx + c

u = x 2
x 2 .
⎛
⎝x 2 +1 ⎞ ⎠ ⎛ ⎝x 2 −5 ⎞ ⎠
x 4 −5x 2 + x 2 −5
x 4 −4x 2 −5✓
h 4 + 4h 2 − 12.
y 4 − y 2 − 20.
(x −2) 2 +7(x −2) +12
(x −2) u = x −2
ax 2 + bx + c
u = x −2
x −2.
(x −2) 2 7(x −2)
(x − 5) 2 + 6(x − 5) + 8.
⎛
⎝y
− 4 ⎞2 ⎠ + 8 ⎛
⎝y
− 4 ⎞ ⎠ + 15.

x 2 + bx + c
x 2 +bx+c.
p 2 +11p +30 w 2 +10x +21 n 2 +19n +48
b 2 +14b +48 a 2 +25a + 100 u 2 + 101u + 100
x 2 −8x +12 q 2 −13q +36 y 2 −18y +45
m 2 −13m +30 x 2 −8x +7 y 2 −5y +6
5p −6+p 2 6n −7+n 2 8−6x + x 2
7x + x 2 +6 x 2 −12−11x −11−10x + x 2
x 2 +bxy+cy 2 .
x 2 −2xy −80y 2 p 2 −8pq −65q 2 m 2 −64mn −65n 2
p 2 −2pq −35q 2 a 2 +5ab −24b 2 r 2 +3rs −28s 2
x 2 −3xy −14y 2 u 2 −8uv −24v 2 m 2 −5mn +30n 2
c 2 −7cd +18d 2
ax 2 + bx + c
p 3 −8p 2 −20p q 3 −5q 2 −24q 3m 3 −21m 2 +30m
11n 3 −55n 2 +44n 5x 4 +10x 3 −75x 2 6y 4 +12y 3 −48y 2
2t 2 +7t +5 5y 2 +16y +11 11x 2 +34x +3
7b 2 +50b +7 4w 2 −5w +1 5x 2 −17x +6
4q 2 −7q −2 10y 2 −53y −11 6p 2 −19pq +10q 2
21m 2 −29mn +10n 2 4a 2 +17ab −15b 2 6u 2 +5uv −14v 2
−16x 2 −32x −16 −81a 2 + 153a +18 −30q 3 − 140q 2 −80q

−5y 3 −30y 2 +35y
ax 2 + bx + c
5n 2 +21n +4 8w 2 +25w +3 4k 2 −16k +15
5s 2 −9s +4 6y 2 + y −15 6p 2 + p −22
2n 2 −27n −45 12z 2 −41z −11 60y 2 + 290y −50
6u 2 −46u −16 48z 3 − 102z 2 −45z 90n 3 +42n 2 − 216n
16s 2 +40s +24 24p 2 + 160p +96 48y 2 +12y −36
30x 2 + 105x −60
x 4 −6x 2 −7 x 4 +2x 2 −8 x 4 −3x 2 −28
x 4 −13x 2 −30 (x −3) 2 −5(x −3) −36 (x −2) 2 −3(x −2) −54
⎛ ⎝3y −2 ⎞ ⎠2 − ⎛ ⎝3y −2 ⎞ ⎠ −2 ⎛ ⎝5y −1 ⎞ ⎠2 −3 ⎛ ⎝5y −1 ⎞ ⎠ −18
u 2 −12u +36 x 2 −14x −32 r 2 −20rs +64s 2
q 2 −29qr −96r 2 12y 2 −29y +14 12x 2 +36y −24z
6n 2 +5n −4 3q 2 +6q +2 13z 2 +39z −26
5r 2 +25r +30 3p 2 +21p 7x 2 −21x
6r 2 +30r +36 18m 2 +15m +3 24n 2 +20n +4
4a 2 +5a +2 x 4 −4x 2 −12 x 4 −7x 2 −8
(x +3) 2 −9(x +3) −36
(x +2) 2 −25(x +2) −54

x 2 + bx + c
(x + m)(x + n).
ax 2 + bx + c.
x 2 − x −20 (x +5)(x −4).
(x +4)(x −5).
(x −5)(x −4).
a
b

⎛ ⎝ ⎛ 3x 2⎞ ⎠ ⎞3 .
(m + 4) 2 .
(x − 3)(x + 3).
9x 2 +24x +16 3x +4.
ax 2 + bx + c.
a 2 +2ab + b 2 = (a + b) 2
a 2 −2ab + b 2 = (a − b) 2
a 2 . b 2 .
2ab?

9x 2 +12x +4.
4x 2 + 12x + 9.
9y 2 + 24y + 16.
a 2 −2ab + b 2 , (a − b) 2 .
Step 1. Does the trinomial fit the pattern? a 2 + 2ab + b 2 a 2 − 2ab + b 2
Is the first term a perfect square? (a) 2 (a) 2
Wr
ite it as a square.
Is the last term a perfect square? (a) 2 (b) 2 (a) 2 (b) 2
Wr
ite it as a square.
Check the middle term. Is it 2ab? (a) 2 ↘ 2 · a · b
↙ (b)2 (a) 2 ↘ 2 · a · b
↙ (b)2
Step 2. Wr
ite the square of the binomial. (a + b) 2 (a − b) 2
Step 3. Check by multiplying.

81y 2 −72y + 16.
(a − b) 2 .
(a − b) 2 ?
⎛
⎝9y −4 ⎞ 2
⎠
(9y) 2 −2·9y ·4+4 2
81y 2 −72y +16✓
64y 2 − 80y + 25.
16z 2 − 72z + 81.
36x 2 +84xy +49y 2 .
⎛
⎝6x +7y ⎞ 2
⎠
(6x) 2 +2·6x ·7y + ⎛ ⎝7y ⎞ 2
⎠
36x 2 +84xy +49y 2 ✓

49x 2 + 84xyx + 36y 2 .
64m 2 + 112mn + 49n 2 .
100x 2 y −80xy +16y.
4y,
4y(5x −2) 2
4y ⎡ ⎣(5x) 2 −2·5x ·2+2 2⎤ ⎦
4y ⎛ ⎝25x 2 −20x +4 ⎞ ⎠
100x 2 y −80xy +16y ✓
8x 2 y − 24xyx + 18y.
27p 2 q + 90pq + 75q.

64y 2 −1.
121m 2 − 1.
81y 2 − 1.
Step 1. Does the binomial fit the pattern? a 2 − b 2
Is this a difference?
____ − ____
Are the first and last terms perfect squares?
Step 2. Wr
ite them as squares. (a) 2 − (b) 2
Step 3. Wr
ite the product of conjugates. (a − b)(a + b)
Step 4. Check by multiplying.
a 2 + b 2

144x 2 −49y 2 .
144x 2 −49y 2
(12x) 2 − ⎛ ⎝7y ⎞ ⎠2
⎛
⎝12x −7y ⎞ ⎠ ⎛ ⎝12x +7y ⎞ ⎠
⎛
⎝12x −7y ⎞ ⎠ ⎛ ⎝12x +7y ⎞ ⎠
144x 2 −49y 2 ✓
196m 2 − 25n 2 .
121p 2 − 9q 2 .
48x 4 y 2 − 243y 2 .

48x 4 y 2 − 243y 2
3y 2
3y 2 ⎛ ⎝16x 4 −81 ⎞ ⎠
3y 2 ⎛⎛ ⎝ 4x 2⎞ 2
⎠ − (9) 2⎞
⎠
3y 2 ⎛ ⎝4x 2 −9 ⎞ ⎠ ⎛ ⎝4x 2 +9 ⎞ ⎠
3y 2 ⎛ ⎝(2x) 2 −(3) 2⎞ ⎠ ⎛ ⎝4x 2 +9 ⎞ ⎠
3y 2 (2x −3)(2x +3) ⎛ ⎝4x 2 +9 ⎞ ⎠
3y 2 (2x −3)(2x +3) ⎛ ⎝4x 2 +9 ⎞ ⎠
3y 2 ⎛ ⎝4x 2 −9 ⎞ ⎠ ⎛ ⎝4x 2 +9 ⎞ ⎠
3y 2 ⎛ ⎝16x 4 −81 ⎞ ⎠
48x 4 y 2 − 243y 2 ✓
2x 4 y 2 − 32y 2 .
7a 4 c 2 − 7b 4 c 2 .
x 2 −6x +9−y 2 .

⎛ ⎝x − y −3 ⎞ ⎠ ⎛ ⎝x + y −3 ⎞ ⎠.
x 2 − 10x + 25 − y 2 .
x 2 + 6x + 9 − 4y 2 .
a 3 + b 3 = (a + b) ⎛ ⎝a 2 − ab + b 2⎞ ⎠
a 3 − b 3 = (a − b) ⎛ ⎝a 2 + ab + b 2⎞ ⎠
a 3 + b 3 = (a + b) ⎛ ⎝a 2 − ab + b 2⎞ ⎠
a 3 − b 3 = (a − b) ⎛ ⎝a 2 + ab + b 2⎞ ⎠

n 3
x 3 + 64.
x 3 + 27.
y 3 + 8.

27u 3 − 125v 3 .
8x 3 − 27y 3 .
1000m 3 − 125n 3 .
6x 3 y +48y 4 .

6y.
500p 3 + 4q 3 .
432c 3 + 686d 3 .
(x +5) 3 −64x 3 .
⎛
⎝y
+ 1 ⎞3 ⎠ − 27y 3 .
(n + 3) 3 − 125n 3 .

16y 2 +24y +9 25v 2 +20v +4 36s 2 +84s +49
49s 2 + 154s + 121 100x 2 −20x +1 64z 2 −16z +1
25n 2 − 120n + 144 4p 2 −52p + 169 49x 2 +28xy +4y 2
25r 2 +60rs +36s 2 100y 2 −20y +1 64m 2 −16m +1
10jk 2 +80jk + 160j 64x 2 y −96xy +36y 75u 4 −30u 3 v +3u 2 v 2
90p 4 + 300p 3 q + 250p 2 q 2
25v 2 −1 169q 2 −1 4−49x 2
121 − 25s 2 6p 2 q 2 −54p 2 98r 3 −72r
24p 2 +54 20b 2 + 140 121x 2 − 144y 2
49x 2 −81y 2 169c 2 −36d 2 36p 2 −49q 2
16z 4 −1 m 4 − n 4 162a 4 b 2 −32b 2
48m 4 n 2 − 243n 2 x 2 −16x +64−y 2 p 2 +14p +49−q 2
a 2 +6a +9−9b 2 m 2 −6m +9−16n 2
x 3 + 125 n 6 + 512 z 6 −27
v 3 − 216 8 − 343t 3 125 − 27w 3
8y 3 − 125z 3 27x 3 −64y 3 216a 3 + 125b 3
27y 3 +8z 3 7k 3 +56 6x 3 −48y 3
2x 2 −16x 2 y 3 −2x 3 y 2 −16y 5 (x +3) 3 +8x 3

(x +4) 3 −27x 3 ⎛ ⎝y −5 ⎞ ⎠3 −64y 3 ⎛ ⎝y −5 ⎞3 ⎠ + 125y 3
64a 2 −25 121x 2 − 144 27q 2 −3
4p 2 − 100 16x 2 −72x +81 36y 2 +12y +1
8p 2 +2 81x 2 + 169 125 − 8y 3
27u 3 + 1000 45n 2 +60n +20 48q 3 −24q 2 +3q
x 2 −10x +25−y 2 x 2 +12x +36−y 2 (x +1) 3 +8x 3
⎛ ⎝y −3 ⎞3 ⎠ −64y 3
n 2 +25≠(n +5) 2 .
y 2 −30y +81 (y −9) 2 .
a
b

x 2 + bx + c?
ax 2 + bx + c?

7x 3 −21x 2 −70x.
7x 3 −21x 2 −70x
7x
7x ⎛ ⎝x 2 −3x −10 ⎞ ⎠
7x(x )(x )
7x(x +2)(x −5)
7x(x +2)(x −5)
7x ⎛ ⎝x 2 −5x +2x −10 ⎞ ⎠
7x ⎛ ⎝x 2 −3x −10 ⎞ ⎠
7x 3 −21x 2 −70x ✓
8y 3 + 16y 2 − 24y.
5y 3 − 15y 2 − 270y.
24y 2 − 150.

24y 2 − 150
6 ⎛ ⎝4y 2 −25 ⎞ ⎠
6 ⎛ ⎝ ⎛ ⎝2y ⎞ ⎠2 − (5) 2⎞ ⎠
6 ⎛ ⎝2y −5 ⎞ ⎠ ⎛ ⎝2y +5 ⎞ ⎠
Is the expression factored completely?
Neither binomial can be factored.
Multiply.
6 ⎛ ⎝2y −5 ⎞ ⎠ ⎛ ⎝2y +5 ⎞ ⎠
6 ⎛ ⎝4y 2 −25 ⎞ ⎠
24y 2 − 150 ✓
16x 3 − 36x.
27y 2 − 48.
4a 2 −12ab +9b 2 .

4a 2 −12ab +9b 2
a ≠1
(2a) 2 −12ab + (3b) 2
a 2 −2ab + b 2 (2a) 2 ↘ −12ab + −2(2a)(3b) ↙ (3b)2
(2a −3b) 2
Is the expression factored completely? Yes.
The binomial cannot be factored.
Multiply.
(2a −3b) 2
(2a) 2 −2·2a ·3b + (3b) 2
4a 2 −12ab +9b 2 ✓
4x 2 + 20xyx + 25y 2 .
9x 2 − 24xyx + 16y 2 .
12x 3 y 2 +75xy 2 .

12x 3 y 2 +75xy 2
3xy 2
3xy 2 ⎛ ⎝4x 2 +25 ⎞ ⎠
Is the expression factored completely? Yes.
Multiply.
3xy 2 ⎛ ⎝4x 2 +25 ⎞ ⎠
12x 3 y 2 +75xy 2 ✓
50x 3 y + 72xyx .
27xyx 3 + 48xyx .
24x 3 +81y 3 .

250m 3 + 432n 3 .
2p 3 + 54q 3 .
3x 5 y −48xy.
3x 5 y −48xy
3xy
3xy ⎛ ⎝x 4 −16 ⎞ ⎠
3xy ⎛ ⎝ ⎛ ⎝x 2⎞ 2
⎠ − (4) 2⎞
⎠
3xy ⎛ ⎝x 2 −4 ⎞ ⎠ ⎛ ⎝x 2 +4 ⎞ ⎠
3xy ⎛ ⎝(x) 2 − (2) 2⎞ ⎠ ⎛ ⎝x 2 +4 ⎞ ⎠
3xy(x −2)(x +2) ⎛ ⎝x 2 +4 ⎞ ⎠
3xy(x −2)(x +2) ⎛ ⎝x 2 +4 ⎞ ⎠
3xy ⎛ ⎝x 2 −4 ⎞ ⎠ ⎛ ⎝x 2 +4 ⎞ ⎠
3xy ⎛ ⎝x 4 −16 ⎞ ⎠
3x 5 y −48xy ✓
4a 5 b − 64ab.
7xyx 5 − 7xyx .
4x 2 +8bx −4ax −8ab.

4x 2 +8bx −4ax −8ab
4 ⎛ ⎝x 2 +2bx − ax −2ab ⎞ ⎠
4 ⎡ ⎣x(x +2b) − a(x +2b) ⎤ ⎦
4(x +2b)(x − a)
4(x +2b)(x − a)
4 ⎛ ⎝x 2 − ax +2bx −2ab ⎞ ⎠
4x 2 +8bx −4ax −8ab ✓
6x 2 − 12xc + 6bx − 12bc.
16x 2 + 24xyx − 4x − 6y.
40x 2 y +44xy −24y.
40x 2 y +44xy −24y
4y
4y ⎛ ⎝10x 2 +11x −6 ⎞ ⎠
a ≠1
4y ⎛ ⎝10x 2 +11x −6 ⎞ ⎠
4y(5x −2)(2x +3)
4y(5x −2)(2x +3)
4y ⎛ ⎝10x 2 +11x −6 ⎞ ⎠
40x 2 y +44xy −24y ✓

4p 2 q − 16pq + 12q.
6pq 2 − 9pq − 6p.
9x 2 −12xy +4y 2 − 49.
9x 2 −12xy +4y 2 −49
9x 2 −12xy +4y 2 −49
a ≠1
(3x) 2 −12xy + ⎛ ⎝2y ⎞ ⎠2 −49
a 2 −2ab + b 2
(3x) 2 ↘ −12xy + ↙ ⎛ ⎝2y ⎞ ⎠2 −49
−2(3x) ⎛ ⎝2y ⎞ ⎠
⎛
⎝3x −2y ⎞ ⎠2 −49
⎛
⎝3x −2y ⎞2 ⎠ −7 2
⎛
⎝ ⎛ ⎝3x −2y ⎞ ⎠ −7 ⎞ ⎠ ⎛ ⎝ ⎛ ⎝3x −2y ⎞ ⎠ +7 ⎞ ⎠
⎛
⎝3x −2y −7 ⎞ ⎠ ⎛ ⎝3x −2y +7 ⎞ ⎠
⎛
⎝3x −2y −7 ⎞ ⎠ ⎛ ⎝3x −2y +7 ⎞ ⎠
9x 2 −6xy −21x −6xy +4y 2 +14y +21x −14y −49
9x 2 −12xy +4y 2 −49✓
4x 2 − 12xyx + 9y 2 − 25.
16x 2 − 24xyx + 9y 2 − 64.

2n 2 +13n −7 8x 2 −9x −3 a 5 +9a 3
75m 3 +12m 121r 2 − s 2 49b 2 −36a 2
8m 2 −32 36q 2 − 100 25w 2 −60w +36
49b 2 − 112b +64 m 2 +14mn +49n 2 64x 2 +16xy + y 2
7b 2 +7b −42 30n 2 +30n +72 3x 4 y −81xy
4x 5 y −32x 2 y k 4 −16 m 4 −81
5x 5 y 2 −80xy 2 48x 5 y 2 − 243xy 2 15pq −15p +12q −12
12ab −6a +10b −5 4x 2 +40x +84 5q 2 −15q −90
4u 5 +4u 2 v 3 5m 4 n + 320mn 4 4c 2 +20cd +81d 2
25x 2 +35xy +49y 2 10m 4 − 6250 3v 4 − 768
36x 2 y +15xy −6y 60x 2 y −75xy +30y 8x 3 −27y 3
64x 3 + 125y 3 y 6 −1 y 6 +1
9x 2 −6xy + y 2 −49 16x 2 −24xy +9y 2 −64 (3x +1) 2 −6(3x +1) +9
(4x −5) 2 −7(4x −5) +12
y 4 − 625
⎛
⎝y 2 −25 ⎞ ⎠ ⎛ ⎝y 2 +25 ⎞ ⎠.

a
b

5y − 3 = 0.
n 3 − 9n 2 − 22n.
f (x) = 8x − 16, f (3) f (x) = 0.
ax + b = c.
x 2 +5x +6=0 3y 2 +4y =10 64u 2 −81=0 n(n +1) =42
n 2 + n.
ax 2 + bx + c =0, a ≠0. a =0, 0·x 2 =0
ax 2 + bx + c =0
a, b, and c are real numbers and a ≠0

a · b =0, a =0 b =0
(5n −2)(6n −1) =0.
(3m − 2)(2m + 1) = 0.
⎛ ⎝ ⎛ 4p + 3 ⎞ ⎠ ⎞⎛ ⎝⎛ 4p − 3 ⎞ ⎠ ⎞ = 0.

ax 2 + bx + c =0.
2y 2 =13y + 45.
3c 2 = 10c − 8.
2d 2 − 5d = 3.
ax 2 + bx + c = 0.

169q 2 = 49.
169x 2 =49
169x 2 −49=0
(13x −7)(13x +7) =0
0
13x −7 = 0 13x +7 = 0
13x = 7 13x = −7
x =
13
7 x = −
13
7
25p 2 = 49.
36x 2 = 121.
(3x −8)(x −1) =3x.
(3x −8)(x −1) =3x
3x 2 −11x +8=3x
3x 2 −14x +8=0
(3x −2)(x −4) =0
3x −2 = 0 x −4 = 0
3x = 2 x = 4
x = 2 3
(2m + 1)(m + 3) = 12m.

(k + 1)(k − 1) = 8.
3x 2 =12x + 63.
3x 2 =12x +63
3x 2 −12x −63=0
3 ⎛ ⎝x 2 −4x −21 ⎞ ⎠ =0
3(x −7)(x +3) =0
3≠0 x −7 = 0 x +3 = 0
3≠0 x = 7 x = −3
18a 2 − 30 = −33a.
123b =−6 − 60b 2 .
9m 3 + 100m =60m 2 .
9m 3 + 100m =60m 2
9m 3 −60m 2 + 100m =0
m ⎛ ⎝9m 2 −60m + 100 ⎞ ⎠ =0
m(3m − 10)(3m − 10) = 0
m =0 3m −10 = 0 3m −10 = 0
m =0 m = 10 3
m = 10 3

8x 3 = 24x 2 − 18x.
16y 2 = 32y 3 + 2y.
f (x) = x 2 +2x −2,
a f (x) =6b
a
f (x) = x 2 +2x −2
f (x)
6=x 2 +2x −2
x 2 +2x −8=0
(x +4)(x −2) =0
x +4 = 0 or x −2 = 0
x = −4 or x = 2
f (x) = x 2 +2x −2 f (x) = x 2 +2x −2
f (−4) = (−4) 2 +2(−4) −2 f (2) =2 2 +2·2−2
f (−4) =16−8−2
f (2) =4+4−2
f (−4) =6✓ f (2) =6✓
b f (−4) =6 f (2) =6, (−4, 6) (2, 6)
f (x) = x 2 − 2x − 8,
a f (x) = 7 b
f (x) = x 2 − 8x + 3,
a f (x) = −4 b

f (x) =0,
f (x) =0,
(x, 0)
f (x) =3x 2 +10x −8,
a b c
a
f (x) =3x 2 +10x −8
f (x)
0=3x 2 +10x −8
(x +4)(3x −2) =0
x + 4 = 0 or 3x −2 = 0
x = −4 or x = 2 3
b y =0. f (−4) =0 f ⎛ ⎝ 2 ⎞
3⎠ =0,
(−4, 0) ⎛ ⎝ 2 3 ,0⎞ ⎠
c x =0. f (0).
f (x) =3x 2 +10x −8
f (0) x
f (0) =3·0 2 +10·0−8
f (0) =−8
f (0) =−8, (0, −8)
f (x) = 2x 2 − 7x + 5,
a b c

f (x) = 6x 2 + 13x − 15,
a b c

n = the first integer.
n + 2 = next consecutive odd integer
n(n +2) = 323
n 2 +2n = 323
n 2 +2n − 323 = 0
(n −17)(n +19) =0
n −17 = 0 n +19 = 0
n = 17 n = −19
n =17
n = −19
n +2 n +2
17 + 2 −19 + 2
19 −17
17, 19 −17, −19
17, 19 and −19, −17.
17 · 19 = 323 ✓ −19(−17) = 323 ✓
−19, −17.

w =
w +4=
A = l · w
117 = (w +4)w
117 = w 2 +4w
117 = w 2 +4w
0=w 2 +4w − 117
0=(w 2 + 13)(w −9)
0=w +13 0=w −9
−13 = w 9=w
w =9
w +4
9+4

⎛ ⎝a 2 + b 2 = c 2⎞ ⎠.
x =
x −7=
a 2 + b 2 = c 2
x 2 +(x −7) 2 =17 2
x 2 + x 2 −14x + 49 = 289
2x 2 −14x + 49 = 289

2x 2 −14x − 240 = 0
2(x 2 −7x − 120) = 0
2(x − 15)(x +8)=0
2≠0 x −15=0 x +8=0
2≠0 x =15 x =−8
x =−8
2≠0 x =15 x =−8
h(t) = −16t 2 +64t +80
a b

c t =2
a h(t) =0.
h(t) =0
h(t). −16t 2 +64t +80=0
−16.
−16(t 2 −4t −5)=0
−16(t −5)(t +1) =0
t −5 = 0 t +1 = 0
t = 5 t = −1
t =5
t =−1
b h(t) = 80.
h(t) =80
h(t). −16t 2 +64t +80=80
−16t.
−16t 2 +64t =0
−16t(t −4)=0
−16t = 0 t −4 = 0
t = 0 t = 4
c t =2 h(2).
h(t) = −16t 2 +64t +80
h(2) t. h(2) = −16(2) 2 +64·2+80
h(2) = 144

h(t) = −16t 2 + 48t + 160
a b
c t =1.5
h(t) = −16t 2 + 32t + 128
a b
c t = 1

(3a −10)(2a −7) =0 (5b +1)(6b +1) =0 6m(12m −5) =0
2x(6x −3) =0 (2x −1) 2 =0 (3y +5) 2 =0
5a 2 −26a =24 4b 2 +7b =−3 4m 2 =17m −15
n 2 =5n −6 7a 2 +14a =7a 12b 2 −15b =−9b
49m 2 = 144 625 = x 2 16y 2 =81
64p 2 = 225 121n 2 =36 100y 2 =9
(x +6)(x −3) =−8 (p −5)(p +3)=−7 (2x +1)(x −3) =−4x
(y −3) ⎛ ⎝y +2 ⎞ ⎠ =4y (3x −2)(x +4) =12x (2y −3) ⎛ ⎝3y −1 ⎞ ⎠ =8y
20x 2 −60x = −45 3y 2 −18y = −27 15x 2 −10x =40
14y 2 −77y = −35 18x 2 − 9 = −21x 16y 2 + 12 = −32y
16p 3 =24p 2 –9p m 3 −2m 2 =−m 2x 3 +72x =24x 2
3y 3 +48y =24y 2 36x 3 +24x 2 =−4x 2y 3 +2y 2 =12y
f (x) = x 2 −8x +8, a
f (x) =−4 b
f (x) =8x 2 −18x +5, a
f (x) =−4 b
f (x) = x 2 +11x + 20, a
f (x) =−8 b
f (x) =18x 2 +15x − 10, a
f (x) =15 b
a b c
f (x) =9x 2 −4
f (x) =25x 2 −49

f (x) =6x 2 −7x −5 f (x) =12x 2 −11x +2
h(t) = −16t 2 +32t
a
b
h(t) = −16t 2 +32t +48
a
b
c
t =1

a
b

x
a(b + c) = ab + ac and ab + ac = a(b + c)
Noun: 7 is a factor of 14
Verb: factor 3 from 3a +3
x 2 + bx + c.
x 2 + bx + c
(x )(x )
multiply to c, m · n = c
add to b, m + n = b
(x + m)(x + n)

x 2 + bx + c
x 2 + bx + c
(x + m)(x + n)
When c is positive, m and n have the same sign.
b positive
b negative
m, n positive m, n negative
x 2 +5x +6 x 2 −6x +8
(x +2)(x +3) (x −4)(x −2)
same signs
same signs
When c is negative, m and n have opposite signs.
x 2 + x −12
x 2 −2x −15
(x +4)(x −3) (x −5)(x +3)
opposite signs
opposite signs
ax 2 + bx + c
ax 2 + bx + c
Multiply to ac. m · n = a · c
Add to b. m + n = b
ax 2 + bx + c
ax 2 + mx + nx + c
a 2 +2ab + b 2 = (a + b) 2
a 2 −2ab + b 2 = (a − b) 2

Step 1. Does the trinomial fit the pattern? a 2 +2ab + b 2 a 2 −2ab + b 2
Is the first term a perfect square? (a) 2 (a) 2
Write it as a square.
Is the last term a perfect square? (a) 2 (b) 2 (a) 2 (b) 2
Write it as a square.
Check the middle term. Is it 2ab? (a) 2 ↘ 2·a · b
↙ (b)2 (a) 2 ↘ 2·a · b
↙ (b)2
Step 2. Write the square of the binomial. (a + b) 2 (a − b) 2
Step 3. Check by multiplying.
a, b
Step 1. Does the binomial fit the pattern? a 2 − b 2
Is this a difference?
____ − ____
Are the first and last terms perfect squares?
Step 2. Write them as squares. (a) 2 − (b) 2
Step 3. Write the product of conjugates. (a − b)(a + b)
Step 4. Check by multiplying.
a 3 + b 3 = (a + b) ⎛ ⎝a 2 − ab + b 2⎞ ⎠
a 3 − b 3 = (a − b) ⎛ ⎝a 2 + ab + b 2⎞ ⎠

x 2 + bx + c?
ax 2 + bx + c?
ax 2 + bx + c =0
a, b, c are real numbers and a ≠0
a · b =0, a =0 b =0
ax 2 + bx + c =0.

f (x) =0,
12a 2 b 3 ,15ab 2 12m 2 n 3 ,42m 5 n 3 15y 3 ,21y 2 ,30y
45x 3 y 2 ,15x 4 y,10x 5 y 3
35y +84 6y 2 +12y −6 18x 3 −15x
15m 4 +6m 2 n 4x 3 −12x 2 +16x −3x +24
−3x 3 +27x 2 −12x 3x(x −1) +5(x −1)
ax − ay + bx − by x 2 y − xy 2 +2x −2y x 2 +7x −3x −21
4x 2 −16x +3x −12 m 3 + m 2 + m +1 5x −5y − y + x
x 2 + bx + c
x 2 +bx+c.
a 2 +14a +33 k 2 −16k +60 m 2 +3m −54
x 2 −3x −10

x 2 +bxy+cy 2 .
x 2 +12xy +35y 2 r 2 +3rs −28s 2 a 2 +4ab −21b 2
p 2 −5pq −36q 2 m 2 −5mn +30n 2
ax 2 + bx + c
x 3 +5x 2 −24x 3y 3 −21y 2 +30y 5x 4 +10x 3 −75x 2
5y 2 +14y +9 8x 2 +25x +3 10y 2 −53y −11
6p 2 −19pq +10q 2 −81a 2 + 153a +18
ax 2 + bx + c
2x 2 +9x +4 18a 2 −9a +1 15p 2 +2p −8
15x 2 +6x −2 8a 2 +32a +24 3x 2 +3x −36
48y 2 +12y −36 18a 2 −57a −21 3n 4 −12n 3 −96n 2
x 4 −13x 2 −30 (x −3) 2 −5(x −3) −36
25x 2 +30x +9 36a 2 −84ab +49b 2 40x 2 + 360x + 810
5k 3 −70k 2 + 245k 75u 4 −30u 3 v +3u 2 v 2
81r 2 −25 169m 2 − n 2 25p 2 −1
9 − 121y 2 20x 2 − 125 169n 3 − n
6p 2 q 2 −54p 2 24p 2 +54 49x 2 −81y 2
16z 4 −1 48m 4 n 2 − 243n 2 a 2 +6a +9−9b 2
x 2 −16x +64−y 2

a 3 − 125 b 3 − 216 2m 3 +54
81m 3 +3
24x 3 +44x 2 24a 4 −9a 3 16n 2 −56mn +49m 2
6a 2 −25a −9 5u 4 −45u 2 n 4 −81
64j 2 + 225 5x 2 +5x −60 b 3 −64
m 3 + 125 2b 2 −2bc +5cb −5c 2 48x 5 y 2 − 243xy 2
5q 2 −15q −90 4u 5 v +4u 2 v 3 10m 4 − 6250
60x 2 y −75xy +30y 16x 2 −24xy +9y 2 −64
(a −3)(a +7) =0 (5b +1)(6b +1) =0 6m(12m −5) =0
(2x −1) 2 =0 3m(2m −5)(m +6) =0
x 2 +9x +20=0 y 2 − y −72=0 2p 2 −11p =40
q 3 +3q 2 +2q =0 144m 2 −25=0 4n 2 =36
(x +6)(x −3) =−8 (3x −2)(x +4) =12x 16p 3 =24p 2 −9p
2y 3 +2y 2 =12y
f (x) = x 2 +11x + 20, a
f (x) =−8 b
f (x) =9x 2 −18x +5, a
f (x) =−3 b
a b c

f (x) =64x 2 −49 f (x) =6x 2 −13x −5
h(t) = −16t 2 +64t +80
a
b
c
t =2

80a 2 + 120a 3 5m(m −1) +3(m −1) x 2 +13x +36
p 2 + pq −12q 2 xy −8y +7x −56 40r 2 + 810
9s 2 −12s +4 6x 2 −11x −10 3x 2 −75y 2
6u 2 +3u −18 x 3 + 125 32x 5 y 2 − 162xy 2
6x 4 −19x 2 +15 3x 3 −36x 2 + 108x
5a 2 +26a =24
h(t) = −16t 2 +64t +80
a
b
c
t =4
f (x) = x 2 −7x +5, a
f (x) =−7 b
f (x) =25x 2 − 81, a
b
c

90y
15y 2.

14
15 · 635 .
12
10 ÷ 8 25 .
q p , q ≠0.
− 24
56
5x
12y
4x +1
x 2 −9
4x 2 +3x −1
2x −8
− 24
56
a 8a2 b
3c
b
4b −3
2b +5 c x +4
x 2 +5x +6 .
a

8a 2 b
3c
3c =0
8a 2 b
3c
c =0
is undefined for c =0.
b
4b −3
2b +5
2b +5 = 0
2b = −5
b = − 5 2
4b −3
2b +5 is undefined for b =−5 2 .
c
x +4
x 2 +5x +6
x 2 +5x +6=0
(x +2)(x +3) =0
x +2=0 or x +3=0
x =−2orx =−3
x +4
x 2 is undefined for x =−2orx =−3.
+5x +6
a 3y2
8x b 8n − 5
3n + 1 c a + 10
a 2 + 4a + 3
a 4p
5q b y − 1
3y + 2 c m − 5
m 2 + m − 6

x +2is simplified because there are no common factors of x + 2 and x +3.
x +3
2x is not simplified because x is a common factor of 2x and 3x.
3x
b ≠0,c ≠0,
then a b = a · c
b · c
and
a · c
b · c = a b .
b ≠0,c ≠0
x +5 x 2+5
2 !
x 2 +5x +6
x 2 +8x +12

x 2 − x − 2
x 2 − 3x + 2 .
x 2 − 3x − 10
x 2 + x − 2 .
3a2 −12ab +12b 2
6a 2 −24b 2
3a 2 −12ab +12b 2
6a 2 −24b 2
3 ⎛ ⎝a 2 −4ab +4b 2⎞ ⎠
6 ⎛ ⎝a 2 −4b 2⎞ ⎠
3(a −2b)(a −2b)
6(a +2b)(a −2b)
a −2b and 3.
3(a −2b)(a −2b)
3 ·2(a +2b)(a −2b)
a −2b
2(a +2b)
2x2 − 12xyx + 18y 2
3x 2 − 27y 2
5x2 − 30xyx + 25y 2
2x 2 − 50y 2
−a −a =−1·a.

7 −7
−1
−a a −1
b − a a − b.
Let’s look at the expression b − a. b − a
Rewrite.
−a + b
Factor out –1. −1(a − b)
a − b b − a. a − b
b − a
−1.
a − b b − a.
−1.
a − b
b − a =−1 a ≠ b
a + b b + a a + b = b + a
a ≠−b a + b
b + a =1.
x 2 −4x −32
64 − x 2 .
x 2 − 4x − 5
25 − x 2 .
x 2 + x − 2
1 − x 2 .
q ≠0,s ≠0,

p
q · rs pr
= qs
x ≠0, x ≠3, x ≠4.
2x
x 2 −7x +12· x2 −9
6x 2 .
5x
x 2 + 5x + 6 · x2 − 4
10x .
9x 2
x 2 + 11x + 30 · x2 − 36
3x 2 .
3a 2 −8a −3
a 2 · a2 +10a +25
−25 3a 2 −14a −5 .

3a 2 −8a −3
a 2 · a2 +10a +25
−25 3a 2 −14a −5
(a −5)(a −5)
(3a +1)(a −3)(a +5)(a +5)
(a −5)(a +5)(3a +1)(a −5)
(3a +1)(a −3)(a +5)(a +5)
(a −5)(a +5) (3a +1)(a −5)
(a −3)(a +5)
(a −5)(a −5)
(a −3)(a +5)
(a −5) 2
2x2 + 5x − 12
x 2 · x 2 − 8x + 16
− 16 2x 2 − 13x + 15 .
4b2 + 7b − 2
1 − b 2 · b 2 − 2b + 1
4b 2 + 15b − 4 .
q ≠0,r ≠0,s ≠0,
p
q ÷ r s = p q · sr
p 3 + q 3
2p 2 +2pq +2q 2 ÷ p2 − q 2
.
6

x 3 − 8
3x 2 − 6x + 12 ÷ x2 − 4.
6
2z 2
z 2 − 1 ÷ z3 − z 2 + z
z 3 .
+ 1
6x 2 −7x +2
4x −8
2x 2 .
−7x +3
x 2 −5x +6

6x 2 −7x +2
4x −8
2x 2 −7x +3
x 2 −5x +6
6x 2 −7x +2
4x −8
÷ 2x2 −7x +3
x 2 −5x +6
6x 2 −7x +2· x 2 −5x +6
4x −8 2x 2 −7x +3
(2x −1)(3x −2)(x −2)(x −3)
4(x −2)(2x −1)(x −3)
(2x −1)(3x −2)(x −2) (x −3)
4(x −2) (2x −1) (x −3)
3x −2
4
3x 2 + 7x + 2
4x + 24
3x 2 .
− 14x − 5
x 2 + x − 30
y 2 − 36
2y 2 + 11y − 6
2y 2 .
− 2y − 60
8y − 4
3x −6 x
4x −4· 2 +2x −3
x 2 ÷ 2x +12
−3x −10 8x +16 .

4m + 4
3m − 15 · m2 − 3m − 10
m 2 ÷ 12m − 36
− 4m − 32 6m − 48 .
2n2 + 10n
n − 1
÷ n2 + 10n + 24
n 2 + 8n − 9 · n + 4
8n 2 + 12n .
q p ,
q ≠0. R(x) = p(x) p(x) q(x)
q(x)
q(x)
R(x) = p(x)
q(x)
p(x) q(x) q(x)
q(x) =0.

R(x) = 2x 2 −14x
4x 2 −16x −48 .
4x 2 −16x −48=0
4 ⎛ ⎝x 2 −4x −12 ⎞ ⎠ =0
4(x −6)(x +2) =0
4≠0 x −6=0 x +2=0
x =6 x =−2
R(x)
x ≠6 x ≠−2.
R(x) =
2x 2 − 10x
4x 2 − 16x − 20 .
R(x) =
4x 2 − 16x
8x 2 − 16x − 64 .
R(x) = f (x) · g(x) f (x) =
2x −6
x 2 −8x +15 g(x) = x2 −25
2x +10 .
R(x) = f (x) · g(x)
R(x) =
R(x) =
R(x) =
R(x) =
R(x) =1
2x −6
x 2 −8x +15· x2 −25
2x +10
2(x −3) (x −5)(x +5)
·
(x −3)(x −5) 2(x +5)
2(x −3)(x −5)(x +5)
2(x −3)(x −5)(x +5)
2 (x −3) (x −5) (x +5)
2 (x −3) (x −5) (x +5)

R(x) = f (x) · g(x) f (x) = 3x − 21
x 2 − 9x + 14 g(x) = 2x2 − 8
3x + 6 .
R(x) = f (x) · g(x) f (x) =
x 2 − x
3x 2 + 27x − 30 g(x) = x2 − 100
x 2 − 10x .
R(x) = f (x) f (x) =
g(x) 3x2
x 2 −4x
g(x) = 9x2
−45x
x 2 −7x +10 .
R(x) = f (x)
g(x)
f (x), g(x). R(x) =
3x 2
x 2 −4x
9x 2 −45x
x 2 −7x +10
f (x) g(x).
R(x) =
R(x) =
3x2
x 2 −4x · x2 −7x +10
9x 2 −45x
3·x · x · (x −5)(x −2)
x(x −4) ·3·3·x · (x −5)
R(x) =
R(x) =
3 · x · x (x −5)(x −2)
x(x −4) ·3·3· x (x −5)
x −2
3(x −4)
R(x) = f (x)
g(x)
f (x) = 2x2
x 2 − 8x g(x) = 8x2 + 24x
x 2 + x − 6 .
R(x) = f (x)
g(x) f (x) = 15x 2
3x 2 + 33x g(x) = 5x − 5
x 2 + 9x − 22 .

a 2x2 4p −1
z b
6p −5 c n −3
n 2 a 10m 6y +13
b
+2n −8
11n 4y −9 c b −8
b 2 −36
a 4x2 y
3y
b 3x −2
2x +1 c
u −1
u 2 −3u −28
a 5pq2
9q
b 7a −4
3a +5 c 1
x 2 −4
− 44
55
56
63 8m 3 n
12mn 2
36v 3 w 2
27vw 3
8n −96
3n −36
12p − 240
5p − 100
x 2 +4x −5
x 2 −2x +1
y 2 +3y −4
y 2 −6y +5
a 2 −4
a 2 +6a −16
y 2 −2y −3
y 2 −9
p 3 +3p 2 +4p +12
p 2 + p −6
x 3 −2x 2 −25x +50
x 2 −25
8b 2 −32b
2b 2 −6b −80
−5c 2 −10c
−10c 2 +30c + 100
3m 2 +30mn +75n 2
4m 2 − 100n 2
5r 2 +30rs −35s 2
r 2 −49s 2
a −5
5−a
5−d
d −5
20 − 5y
y 2 −16
4v −32
64 − v 2 w 3 + 216
w 2 −36
v 3 + 125
v 2 −25
z 2 −9z +20
16 − z 2 a 2 −5a −36
81 − a 2
12
16 · 410
32
5 · 16
24
5x 2 y 4
12xy 3 · 6x2
20y 2
12a 3 b
b 2 · 2ab2
9b 3

5p 2
p 2 −5p −36· p2 −16
10p
3q 2
q 2 + q −6· q2 −9
9q
2y 2 −10y
y 2 +10y +25· y +5
6y
z 2 +3z
z 2 −3z −4· z −4
z 2
28 − 4b
3b −3 · b2 +8b −9
b 2 −49
3c 2 −16c +5
c 2 · c2 +10c +25
−25 3c 2 −14c −5
6m 2 −13m +2
9−m 2 · m 2 −6m +9
6m 2 +23m −4
72m −12m 2 ·
8m +32
m2 +10m +24
m 2 −36
2d 2 + d −3
d 2 · d2 −8d +16
−16 2d 2 −9d −18
2n 2 −3n −14
25 − n 2 · n2 −10n +25
2n 2 −13n +21
v −5
11 − v ÷ v2 −25
v −11
3s 2
s 2 −16 ÷ s3 +4s 2 +16s
s 3 −64
p 3 + q 3
3p 2 +3pq +3q 2 ÷ p2 − q 2
12
10 + w
w −8
r 2 −9
15
÷ 100 − w2
8−w
÷ r 3 −27
5r 2 +15r +45
v 3 −8w 3
2v 2 +4vw +8w 2 ÷ v2 −4w 2
4
x 2 +3x −10
4x
÷(2x 2 +20x + 50)
2y 2 −10yz −48z 2
2y −1
÷(4y 2 −32yz)
2a 2 − a −21
5a +20
a 2 +7a +12
a 2 +8a +16
3b 2 +2b −8
12b +18
3b 2 +2b −8
2b 2 −7b −15
12c 2 −12
2c 2 −3c +1
4c +4
6c 2 −13c +5
4d 2 +7d −2
35d +10
d 2 −4
7d 2 −12d −4
10m 2 +80m ·
3m −9
m2 +4m −21
m 2 −9m +20 ÷ 5m2 +10m
2m −10
4n 2 +32n
3n +2 · 3n2 − n −2
n 2 + n −30 ÷ 108n2 −24n
n +6
12p 2 +3p
p +3
÷ p2 +2p −63
p 2 − p −12 · p −7
9p 3 −9p 2
6q +3
9q 2 −9q ÷ q2 +14q +33
q 2 +4q −5 · 4q2 +12q
12q +6

R(x) = x3 −2x 2 −25x +50
x 2 −25
R(x) = x3 +3x 2 −4x −12
x 2 −4
R(x) = 3x2 +15x
6x 2 +6x −36
R(x) =
8x2 −32x
2x 2 −6x −80
R(x) =f(x) ·g(x) f (x) g(x)
f (x) =
g(x) =
6x2 −12x
x 2 +7x −18
x2 −81
3x 2 −27x
f (x) =
x 2 −2x
x 2 +6x −16
g(x) = x2 −64
x 2 −8x
f (x) = 4x
x 2 f (x) =
−3x −10
2x2 +8x
x 2 −9x +20
g(x) = x2 −25
8x 2 g(x) = x −5
x 2
R(x) = f (x)
g(x)
f (x) g(x)
f (x) =
27x2
3x −21
g(x) = 3x2 +18x
x 2 +13x +42
f (x) = 24x2
2x −8
g(x) = 4x3 +28x 2
x 2 +11x +28
f (x) =
g(x) =
16x2
4x +36
4x2 −24x
x 2 +4x −45
f (x) = 24x2
2x −4
g(x) = 12x2 +36x
x 2 −11x +18
x2 − x −20
x 2
−4
p2 +4p −21
9−p 2 .
a 7 · b
4
910
n 9 c
n −3· n +3
b n =7
a
a
x2 −1
x
24 5
÷6 b
÷(x +1) c
b x =5.
a

a
b

7
10
+ 8 15 .
3x
4 − 8 9 .
6(2x + 1) − 4(x − 5).
r ≠0,
p
r + q r = p + q
r
and
p
r − q r = p − q
r
11x +28
x +4
+ x2
x +4 .
x +4, x =−4.

11x +28+ x +4 x x2
+4 ,
x ≠−4
11x +28+x 2
x +4
x 2 +11x +28
x +4
(x +4)(x +7)
x +4
(x +4)(x +7)
x +4
x +7
x +7 x +4 x ≠−4.
9x + 14
x + 7 + x2
x + 7 .
x 2 + 8x
x + 5 + 15
x + 5 .
5x2 −7x +3
x 2 −3x +18 − 4x2 + x −9
x 2 −3x +18 .

5x 2 −7x +3
x 2 −3x +18 − 4x2 + x −9
x 2 −3x +18
5x 2 −7x +3− ⎛ ⎝4x 2 + x −9 ⎞ ⎠
x 2 −3x +18
5x 2 −7x +3−4x 2 − x +9
x 2 −3x −18
x 2 −8x +12
x 2 −3x −18
(x −2)(x −6)
(x +3)(x −6)
(x −2)(x −6)
(x +3)(x −6)
(x −2)
(x +3)
4x2 − 11x + 8
x 2 − 3x + 2 − 3x2 + x − 3
x 2 − 3x + 2 .
6x2 − x + 20
x 2 − 81
− 5x2 + 11x − 7
x 2 .
− 81
−1
−1 .
−1
−1 .

m2 −6m
m 2 −1
− 3m +2
1−m 2.
−1
−1 .
y 2 − 5y
y 2 − 4 − 6y − 6
4 − y 2.
2n2 + 8n − 1
n 2 − 1
− n2 − 7n − 1
1 − n 2 .
7 + 5 .
12 18

x
a 8
x 2 −2x −3 ,
a
3x
x 2 +4x +3
b
8
x 2 −2x −3 ,
3x
x 2 +4x +3 .
b
a 2
x 2 − x − 12 , 1
x 2 − 16
b
a
3x
x 2 − 3x – 10 , 5
x 2 + 3x + 2
b

3
x −3 + x −2 2 .
2
x − 2 + 5
x + 3 .
4
m + 3 + 3
m + 4 .
3x −6
(x −3)(x −2) 2x −6 .
(x −2)(x −3)

8
x 2 −2x −3 + 3x
x 2 +4x +3 .
Find the LCD.
x 2 −2x −3=(x +1)(x −3)
______________________________
x 2 +4x +3=(x +1) (x +3)
LCD = (x +1)(x −3)(x +3)
1
m 2 − m − 2 + 5m
m 2 + 3m + 2 .
2n
n 2 − 3n − 10 + 6
n 2 + 5n + 6 .
8y
y 2 −16 − y −4 4 .

Find the LCD.
y 2 −16= ⎛ ⎝y −4 ⎞ ⎠ ⎛ ⎝y +4 ⎞ ⎠
____________
y −4 =y −4
LCD = ⎛ ⎝y −4 ⎞ ⎠ ⎛ ⎝y +4 ⎞ ⎠
2x
x 2 − 4 − 1
x + 2 .
3
z + 3 − 6z
z 2 − 9 .
−3n −9
n 2 + n −6 − n 2−n +3.

n −2 2−n
−1
−1 .
a −(−b) =a + b.
Find the LCD.
n 2 + n −6=(n −2)(n +3)
n −2=(n −2)
_________________
LCD = (n −2)(n +3)
3x − 1
x 2 − 5x − 6 − 2
6 − x .
−2y − 2
y 2 + 2y − 8 − y − 1
2 − y .
4
a 2 +6a +5 − 3
a 2 +7a +10 .

Find the LCD.
a 2 +6a +5 =(a +1)(a +5)
a 2 ____________________________
+7a +10 = (a +5)(a +2)
LCD = (a +1)(a +5)(a +2)
3
b 2 − 4b − 5 − 2
b 2 − 6b + 5 .
4
x 2 − 4 − 3
x 2 − x − 2 .
2u
u −1 + 1 u − 2u −1
u 2 − u .

Find the LCD.
u −1 = (u −1)
u = u
_______________
u 2 − u = u(u −1)
LCD = u(u −1)
v
v + 1 + 3
v − 1 − 6
v 2 − 1 .
3w
w + 2 + 2
w + 7 − 17w + 4
w 2 + 9w + 14 .
R(x) = f (x) − g(x) f (x) = x +5
x −2
g(x) = 5x +18
x 2 −4 .
f (x), g(x).
Find the LCD.
x −2 =(x −2)
___________________
x 2 −4=(x −2)(x +2)
LCD = (x −2)(x +2)

R(x) = f (x) − g(x) f (x) = x + 1
x + 3 g(x) = x + 17
x 2 − x − 12 .
R(x) = f (x) + g(x) f (x) = x − 4
x + 3
g(x) = 4x + 6
x 2 − 9 .

2 + 7 7 + 15 15
11
24 24
7m
2m + n + 4
2m + n 2r 2 + 15r −8
2r −1 2r −1
3c
4c −5 + 4c 5
−5
3s 2 + 13s −10
3s −2 3s −2
2w 2
w 2 −16 + 8w
w 2 −16
7x 2
x 2 −9 + 21x
x 2 −9
9a 2
3a −7 − 3a 49
−7
25b 2
5b −6 − 36
5b −6
3m 2 − 21m −30
6m −30 6m −30
2n 2 − 18n −16
4n −32 4n −32
6p 2 +3p +4
p 2 +4p −5 − 5p2 + p +7
p 2 +4p −5
5q 2 +3q −9
q 2 +6q +8 − 4q2 +9q +7
q 2 +6q +8
5r 2 +7r −33
r 2 − 4r2 +5r +30
−49 r 2 −49
7t 2 − t −4
t 2 −25
− 6t2 +12t −44
t 2 −25
10v + 2v +4
2v −1 1−2v
20w + 5w +6
5w −2 2−5w
10x 2 +16x −7+ 8x −3
2x2 +3x −1
3−8x
6y 2 +2y −11
+ 3y2 −3y +17
3y −7 7−3y
z 2 +6z
z 2 −
3z +20
−25 25 − z 2 a 2 +3a
a 2 − 3a −27
−9 9−a 2
2b 2 +30b −13
b 2 − 2b2 −5b −8 c 2 +5c −10
−49 49 − b 2 c 2 −16
− c2 −8c −10
16 − c 2
a b
5
x 2 −2x −8 ,
2x
x 2 − x −12
8
y 2 +12y +35 ,
3y
y 2 + y −42
9
z 2 +2z −8 ,
4z
z 2 −4

6
a 2 +14a +45 , 5a
a 2 −81
4
b 2 +6b +9 , 2b
b 2 −2b −15
5
c 2 −4c +4 , 3c
c 2 −7c +10
2
3d 2 +14d −5 , 5d
3d 2 −19d +6
3
5m 2 −3m −2 , 6m
5m 2 +17m +6
7
10x 2 y + 4
15xy 2 1
12a 3 b 2 + 5
9a 2 b 3 3
r +4 + r −5
2
4
s −7 + 5
s +3
2y
y +3 + 3
y −1
4
cd +3c + 1
d 2 −9
3r
r 2 +7r +6 + 9
r 2 +4r +3
x −3
x +6 − x +3
x
6
m +6 − 12m
m 2 −36
5
3w −2 + 2
w +1
3z
z −2 + 1
z +5
−3m
3m −3 + 5m
m 2 +3m −4
2s
s 2 +2s −8 + 4
s 2 +3s −10
5a
a +3 − a a +2
+6
4
n +4 − 8n
n 2 −16
4
2x +5 + 2
x −1
5b
a 2 b −2a 2 + 2b
b 2 −4
8
4n +4 + 6
n 2 − n −2
t
t −6 − t t −2
+6
3b
b −2 − b −6
b −8
−9p −17
p 2 −4p −21 − p +1
7−p
−13q −8
q 2 +2q −24 − q +2
4−q
−2r −16
r 2 +6r −16 − 2−r
5
2t −30
t 2 +6t −27 − 2
3−t
2x +7
10x −1 +3
8y −4
5y +2 −6 3
x 2 −3x −4 − 2
x 2 −5x +4
4
x 2 −6x +5 − 3
x 2 −7x +10
5a
a −2 + 9 a − 2a +18
a 2 −2a
6d
d −5 + d +4 1 − 7d −5
d 2 − d −20
5
x 2 +8x −9 − 4
x 2 +10x +9
2b
b −5 + 2b 3 − 2b −15
2b 2 −10b
3d
d +2 + d 4 − d +8
d 2 +2d
3
2x 2 +5x +2 − 1
2x 2 +3x +1
c
c +2 + c −2 5 − 10c
c 2 −4
2q
q +5 + q −3 3 13q +15
−
q 2 +2q −15

a R(x) =f(x) +g(x) b R(x) =f(x) −g(x).
f (x) = −5x −5
x 2 + x −6
g(x) =
2−x
x +1
f (x) = 5
x +7
g(x) =
10x
x 2 −49
f (x) = −4x −24
x 2 + x −30
g(x) =
5−x
x +7
f (x) = 6x
x 2 −64
g(x) =
x −8
3
3 x + 4 x 7
2x
.
x 2 +5x +4 x 2 − 16.
1 x + 1 y 2
x + y . a 4
n 2 +6n +9 − 1
n 2 −9
1 x + 1 y . b
2
x + y
a c d
1 x + 1 y .
a
b

3
5 .
9
10
1 − 1
3
4 2 + 4 · 5 .
1
2x
+ 1 4 = 1 8 .
3
4
5
8
x
2
xy
6
4
y −3
8
y 2 −9
1 x + 1 y
x
y − y x
2
x +6
4
x −6 − 4
x 2 −36
6x 2 −7x +2
4x −8
2x 2 −8x +3
x 2 −5x +6

⎛
⎝ 6x2 −7x +2⎞
4x −8 ⎠ ÷ ⎛ ⎝ 2x2 −8x +3⎞
x 2 −5x +6⎠ .
6
x −4 .
3
x 2 −16
6
x −4
3
x 2 −16
6
x −4 ÷ 3
x 2 −16
6
x −4· x2 −16
3
3·2 (x −4)(x +4)
x −4· 3
3·2(x −4)(x +4)
3(x −4)
3 ·2(x −4)(x +4)
3 (x −4)
2(x +4)
x −4 x 2 − 16. x =4 x =−4.
2
x 2 − 1 .
3
x + 1
1
x 2 − 7x + 12 .
2
x − 4

1
3 + 1 6
1
2 − 1 .
3
1
2 + 2 3
5
6 + 1 .
12
3
4 − 1 3
1
8 + 5 .
6
1 x + 1 y
x
y − y .
x

1 x + 1 y
1 x − 1 .
y
1
a + 1 b
1
a 2 − 1
b 2

n − 4n
n +5
1
n +5 + 1 .
n −5
b − 3b
b + 5
2
b + 5 + 1 .
b − 5
1 − 3
c + 4
1
c + 4 + c .
3
LCD
LCD

1
3 + 1 6
1
2 − 1 .
3
1
2 + 1 5
1
10 + 1 .
5
1
4 + 3 8
1
2 − 5 .
16
1 x + 1 y
x
y − y .
x

1
a + 1
b
a
b + b .
a
1
x 2 − 1
y 2
1 x + 1 .
y
2
x +6
4
x −6 − 4 .
x 2 −36
x 2 −36= ⎛ ⎝x +6 ⎞ ⎠ ⎛ ⎝x −6 ⎞ ⎠

3
x + 2
5
x − 2 − 3 .
x 2 − 4
2
x − 7 − x + 1
7
6
x + 7 − 1 .
x 2 − 49
4
m 2 −7m +12
3
m −3 − 2 .
m −4

(m −3)(m −4).
3
x 2 + 7x + 10
4
x + 2 + 1 .
x + 5
4y
y + 5 + 2
y + 6
.
3y
y 2 + 11y + 30
y
y +1
1+ 1 .
y −1
⎛ ⎝y +1 ⎞ ⎠ ⎛ ⎝y −1 ⎞ ⎠.

x
x + 3
1 + 1 .
x + 3
1 + 1
x − 1 .
3
x + 1

2a
a +4
4a 2
a 2 −16
3b
b −5
b 2
b 2 −25
5
c 2 +5c −14
10
c +7
8
d 2 +9d +18
12
d +6
1
2 + 5 6
2
3 + 7 9
1
2 + 3 4
3
5 + 7 10
2
3 − 1 9
3
4 + 5 6
1
2 − 1 6
2
3 + 3 4
n
m + 1 n
1
n − n m
1 p + p q
q
p − 1 q
1 r + 1
t
1
r 2 − 1
t 2
2
v + w
2
1
v 2 − 1
w 2
x − 2x
x +3
1
x +3 + 1
x −3
y − 2y
y −4
2
y −4 + 2
y +4
2− 2
a +3
1
a +3 + a 2
4+ 4
b −5
1
b −5 + b 4
1
3 + 1 8
1
4 + 1 12
1
4 + 1 9
1
6 + 1 12
5
6 + 2 9
7
18 − 1 3
1
6 + 4 15
3
5 − 1 2
c
d + 1 d
1
d − d c
1
m + m n
n
m − 1 n
1 p + 1 q
1
p 2 − 1
q 2
2 r + 2
t
1
r 2 − 1
t 2
2
x +5
3
x −5 + 1
x 2 −25
5
y −4
3
y +4 + 2
y 2 −16
5
z 2 + 3
−64
z +8
1
z +8 + 2
z −8
3
s +6 + 5
s −6
1
s 2 + 4
−36
s +6
4
a 2 −2a −15
1
a −5 + 2
a +3
5
b 2 −6b −27
3
b −9 + 1
b +3
5
c +2 − 3
c +7
5c
c 2 +9c +14

6
d −4 − 2
d +7
2d
d 2 +3d −28
2+ 1
p −3
5
p −3
n
n −2
3+ 5
n −2
m
m +5
4+ 1
m −5
7+ 2
q −2
1
q +2
3
4 − 2 7
1
2 + 5 14
v
w + 1 v
1 v − v w
2
a +4
1
a 2 −16
3
b 2 −3b −40
5
b +5 − 2
b −8
x − 3x
x +2
3
x +2 + 3
x −2
3
m + 3 n
1
m 2 − 1
n 2
y
y +3
2+ 1
y −3
2
r −9
1
r +9 + 3
r 2 −81
3
x +2
x
x 2 −4
1
a + 1
b
1
a − 1
b
1
a + 1 b
1
a − 1 .
b
a
b

1 6 x + 1 2 = 1 3 .
n 2 − 5n − 36 = 0.
5x + 2y =10 y.
Rational Expression Rational Equation
1
8 x + 1 2
y +6
y 2 −36
1
n −3 + n +4
1
1
8 x + 1 2 = 1 4
y +6
y 2 −36 = y +1
1
n −3 + n +4 1 15
n 2 + n −12

x ≠ c
1 x + 1 3 = 5 6 .
1 y + 2 3 = 1 5 .
2 3 + 1 5 = 1 x .

1− 5 y =−6 y 2.

y =2, y =3.
1 − 2 x = 15
x 2.
1 − 4 y = 12
y 2.
2
x +2 + x −2 4 = x −1
x 2 −4 .
(x +2)(x −2).

x =−1.
2
x + 1 + 1
x − 1 = 1
x 2 − 1 .
5
y + 3 + 2
y − 3 = 5
y 2 − 9 .
m +11
m 2 −5m +4 = 5
m −4 − 3
m −1 .

(m −4)(m −1).
x + 13
x 2 − 7x + 10 = 6
x − 5 − 4
x − 2 .
y − 6
y 2 + 3y − 4 = 2
y +4 + 7
y − 1 .
y
y +6 = 72
y 2 −36 +4.

(y −6)(y +6).
y =4.
x
x + 4 = 32
x 2 − 16 + 5.
y
y + 8 = 128
y 2 − 64 + 9.

x
2x −2 − 2
3x +3 = 5x2 −2x +9
12x 2 −12 .
12(x −1)(x +1).
x =1 x =−1
y
5y − 10 − 3y 5
+ 6 = 2y2 − 19y + 54
15y 2 .
− 60
z
2z + 8 − 3
4z − 8 = 3z2 − 16z − 16
8z 2 + 16z − 64 .
4
3x 2 −10x +3 + 3
3x 2 +2x −1 = 2
x 2 −2x −3 .

x ≠−1,x ≠ 1 3 , x ≠3
(3x −1)(x +1)(x −3).
x =3,
x =3
15
x 2 + x − 6 − 3
x − 2 = 2
x + 3 .
5
x 2 + 2x − 3 − 3
x 2 + x − 2 = 1
x 2 + 5x + 6 .
f (x) = 2x −6
x 2 , a b f (x) =1, c
−8x +15
a

x 2 −8x +15=0
(x −3)(x −5) =0
x −3=0 x −5=0
x =3 x =5
x ≠3,x ≠5.
b
(x −3)(x −5).
c x =7,x =3. f (x) =1,
(7, 1), (3, 1).
f (x) = 8 − x
x 2 , a b f (x) = 3 c
− 7x + 12
f (x) = x − 1
x 2 , a b f (x) = 4 c
− 6x + 5

m = y − y 1
x − x 1
Multiply both sides of the equation by x − x 1 . m(x − x 1 ) = ⎛ − y 1 ⎞
⎝y
x − x 1⎠ (x − x 1 )
Simplify. m(x − x 1 ) = y − y 1
Rewrite the equation with the y terms on the left. y − y 1 = m(x − x 1 )
(2, 3).
m = y −2 y.
x −3
x −3.
m = y − 5 y.
x − 4
m = y − 1
x + 5
y.
1 c + m 1 =1

m −1.
c =0,m =0
m ≠1.
1 a + 1 = c
b
2 x + 1 3 = 1 y

1 a + 2 5 = 1 2
6
3
− 2 d = 4 9
4
5
+ 1 4 = 2 v 3
8
+ 2 y = 1 4
1− 2 m = 8 m 2 1+ 4 n = 21
n 2
1+ 9 p = −20
p 2
5
3v −2 = 7 4v
3
x +4 + 7
x −4 = 8
x 2 −16
8
z −10 − 7
z +10 = 5
z 2 − 100
1− 7 q = −6
q 2
8
2w +1 = 3 w
5
y −9 + 1
y +9 = 18
y 2 −81
9
a +11 − 6
a −11 = 6
a 2 − 121
−10
q −2 − 7
q +4 =1 2
s +7 − 3
s −3 =1
v −10
v 2 −5v +4 = 3
v −1 − 6
v −4
x −10
x 2 +8x +12 = 3
x +2 + 4
x +6
b +3
3b
+ b
24 = 1 b
w +8
w 2 −11w +28 = 5
w −7 + 2
w −4
y −5
y 2 −4y −5 = 1
y +1 + 1
y −5
c +3
12c + c
36 = 1 4c
d
d +3 = 18
d 2 −9 +4 m
m +5 = 50
m 2 −25 +6
n
n +2 −3= 8
n 2 −4
p
p +7 −8= 98
p 2 −49
q
3q −9 − 3
4q +12 = 7q2 +6q +63
24q 2 − 216
r
3r −15 − 1
4r +20 = 3r2 +17r +40
12r 2 − 300
s
2s +6 − 5s 2
+5 = 5s2 −3s −7
10s 2 +40s +30
t
6t −12 − 5
2t +10 = t2 −23t +70
12t 2 +36t − 120

2
x 2 +2x −8 − 1
x 2 +9x +20 = 4
x 2 +3x −10
3
x 2 −5x −6 + 3
x 2 −7x +6 = 6
x 2 −1
5
x 2 +4x +3 + 2
x 2 + x −6 = 3
x 2 − x −2
2
x 2 +2x −3 + 3
x 2 +4x +3 = 6
x 2 −1
f (x) = x −2
x 2 +6x +8 ,
b
a
f (x) =5 c
f (x) = 2−x
x 2 −7x +10 ,
b
a
f (x) =2 c
f (x) = x +1
x 2 −2x −3 ,
b
a
f (x) =1 c
f (x) =
a
b f (x) =3
5−x
x 2 +5x +6 ,
c
C r =2π r. I r = P r. v +3
w −1 = 1 w.
2
x +5
2−y = 4 3
y. a = b +3
c −2 c. m = n
2−n
n.
1 p + 2 q =4 p. 3 s + 1 t =2 s. 2 v + 1 5 = 3 w
w.
6 x + 2 3 = 1 y y. m +3
n −2 = 4 5 n. r = s
3−t
t.
E c = m 2 c. R
T
= W T. 3 x − 5 y = 1 4
y.
c = 2 a + b 5
a.
y
y +6 = 72
y 2 −36 +4
y =−6 y =4.

a
b 1 − 10,

2(n −1)− 4 = −10.
1 3 x + 1 4 x = 5 6 .
a b = d c , b ≠0,d ≠0.
1 2 = 4 8 1 2 = 4 8
n
n +14 = 5 7 .

y
y + 55 = 3 8 .
z
z − 84 = − 1 5 .
a
b
= c d ,

ΔABC
ΔXYZ.
ΔABC ΔXYZ,

D = rt

D = rt
t = D r .
D = r · t, t t = D r .
200
r −30 = 300
r +30
(r + 30)(r − 30) ⎛ ⎝r 200 ⎞
−30⎠ =(r + 30)(r − 30)⎛ ⎝r 300 ⎞
+30⎠
(r + 30)(200) = (r − 30)300
200r + 6000 = 300r − 9000
15000 = 100r

Tailwind 150 + 30 = 180 mph 300
180 = 5 3 hours
Headwind 150 − 30 = 120 mph 200
120 = 5 3 hours
D = r · t, t t = D r .
8 r +
r 24
+4 = 3

⎛ ⎝ r +4⎞ ⎠ ⎛ ⎝ 8 r + r 24 ⎞
+4⎠ = 3·r(r +4)
8(r +4)+24r = 3r(r +4)
8r +32+24r = 3r 2 +12r
32 + 32r = 3r 2 +12r
0 = 3r 2 −20r −32
0 = (3r +4)(r −8)
(3r +4)=0 (r −8)=0
r =− 4 3
r =8
r =8
r +4, 8 + 4 = 12.
Run 8 mph 8 miles = 1 hour
8mph
Bike 12 mph 24 miles = 2 hours
12 mph
t.

D = r · t, t t = D r .
12
h −8 = 12 h +2
h(h −8) ⎛ ⎝h 12 ⎞
−8⎠ = h(h −8)⎛ ⎝ 12 h +2⎞ ⎠
12h = 12(h −8)+2h(h −8)
12h = 12h −96+2h 2 −16h
0 = 2h 2 −16h −96
0 = 2(h 2 −8h − 48)
0 = 2(h − 12)(h +4)
h −12 = 0 h +4 = 0
h =12 h =−4
Downhill 12 mph 12 miles = 1 hour
12 mph
Uphill 12 − 8 = 4 mph 12 miles = 3 hours.
4 mph

1 6
1 12
1
t
t.
1 6

1
10
1 8
1 10 1 8 1 t
1 t

1 7
1 s

s =15h
y = kx, where k ≠0
a b
a
c = 315 m =18

a b
a b
y = k x .
y = k x , where k ≠0
1 x .

a b
a
b
f and L. f = 11,440
L
L. f = 11,440
20
f = 572
f L = 20.

a b
p.
a b

x = 7 56
56 8
72 = y 9
72
156 = −6 q a
a +12 = 4 7
98
154 = −7 p
b
b −16 = 11 9
m +90
25
= m +30
15
n +10
4
= 40 − n
6
2p +4
8
= p +18
6
q −2
2
=
2q −7
18
a
b
a
b
1
2

a b
a b

y x
y = 14, x =3.
x y.
a b
a = 16, b =4.
a b.
p q
p = 9.6, q =3.
p q.
v w
v =8, w = 1 .
2
v w.
P,
g,
a
P g. b
d,
v,
a
d v. b
a
b
a
b
a
b

a
b
y x
y =5 x =4,
x y.
p q
p =2 q =1
p q.
v w
v =6 w = 1 2 ,
v
w.
a b
a =12 b = 1 3 ,
a
b.
a
b
a
b
a
b
a
b
a
b
a
b

a
m. b
a
h,
s. b
144 a = 9
4
4 · 144 = 9 · a.
1 3 + 1 4 = 1
t
a
b

x − 5 a x = 6 b x = −3 c x = 5.
8 − 2x < 12.
−3 ≤ x < 5.
2x 3 >1, 2x
x −3 3,

x −1
x +3 ≥0.
x −1
x +3 ≥0
x −1=0 x =1, 1
x +3=0 x =−3,
−3.
−3
(−∞, −3) (−3, 1) (1, ∞)
Interval −∞, −3
−4 (−∞, −3). x =−4
x −1 x +3
(−∞, −3).
(−3, 1). x =0.
Interval −3, 1
x −1 x +3

(−3, 1).
(1, ∞). x =2.
Interval 1, ∞
x −1 x +3
(1, ∞).
(−∞, −3) (1, ∞)
x =−3
x =1
∪,
(−∞, −3) ∪ [1, ∞).
x − 2
x + 4 ≥ 0.
x + 2
x − 4 ≥ 0.

4x
x −6

3x
x − 3 < 1.
3x
x − 4 < 2.
5
x 2 −2x −15 >0.
5
x 2 −2x −15 >0
5
(x +3)(x −5) >0

(x +3)(x −5)=0
x =−3,x =5
(−∞, −3)∪(5, ∞)
1
x 2 + 2x − 8 > 0.
3
x 2 + x − 12 > 0.
1 − 2 3 x 2 < 3x 5 .
1
3 − 2 x 2 < 5 3x
5
3x
1
3
− 2 x 2 − 5 3x

3x 2 = 0 x −6 = 0 x +1 = 0
x = 0 x = 6 x = −1
(−1, 0) (0, 6).
(−1,0)∪(0,6)
1
2
+ 4 x 2 < 3 x .
1
3
+ 6 x 2 < 3 x .
R(x) = x x +3 ,
−5

R(x) ≤0
R(x). x +3
x −5 ≤0
x ≠5
x +3 = 0 x −5 = 0
x = −3 x = 5
⎡
⎣−3, 5)
R(x) = x x − + 2 ,
4
R(x) = x x + − 1 ,
4
C(x)
C(x) x. c(x) = C(x)
x .
C(x) =10x + 3000 x, a
c(x) b
a

C(x) =10x + 3000
c(x) = C(x)
x .
x.
c(x) = C(x)
x
c(x) = 10x + 3000
x
c(x) =
10x + 3000
x .
b
c(x) 40. c(x)

x −3
x +4 ≥0 x +6
x −5 ≥0 x +1
x −3 ≤0
x −4
x +2 ≤0
x −7
x −1 >0 x +8
x +3 >0
x −6
x +5 0
3
x 2 −5x +4 4 3x
1
2
− 4 x 2 ≤ 1 x
1
2
− 3
2x 2 ≥ 1 x 1
x 2 −16 0
4
x −2 ≥ 3
x +1
5
x −1 ≤ 4
x +2
R(x) = x x −5
−2 , R(x) = x x +1
+3 , R(x) = x −6
x +2
x
x
R(x) = x x +1
−4 ,

(−∞, −2] ∪ [4, ∞).
a
b

q p , q ≠0.
R(x) = p(x)
q(x)
q(x)
p(x) q(x)
b ≠0,c ≠0, a b = a · c
b · c a · c
b · c = a b .
a − b b − a.
a − b
b − a =−1
a ≠ b
−1.
q ≠0,s ≠0,
p
q · rs pr
= qs
q ≠0,r ≠0,s ≠0,
p
q ÷ r s = p q · sr

≠0,
p
r + q r = p + q
r
p r − q r = p − q
r

a b = d c , b ≠0,d ≠0.
ΔABC ΔXYZ,
y = kx, k ≠0.
x y.
y = k x , k ≠0.
5a +3
3a −2
b −7
b 2 −25
5x 2 y 2
8y

x −3
x 2 − x −30
18
24 9m 4
18mn 3 x 2 +7x +12
x 2 +8x +16
7v −35
25 − v 2
5
8 · 415
3xy 2
8y 3 · 16y2
24x
72x −12x 2
8x +32 · x2 +10x +24
x 2 −36
2y 2 + y −3
4−y 2 ·
y 2 −4y +4
2y 2 +11y +12
x 2 −4x −12
x 2 +8x +12 ÷ x2 −36
3x
y 2 −16
4
÷
y 3 −64
2y 2 +8y +32
11 + w
w −9
÷ 121 − w2
9−w
3y 2 −12y −63
÷(6y
4y +3
2 −42y)
c 2 −64
3c 2 +26c +16
c 2 −4c −32
15c +10
8a 2 +16a ·
a −4
a2 +2a −24
a 2 +7a +10 ÷ 2a2 −6a
a +5
R(x) = f (x) · g(x)
f (x) = 9x2 +9x
x 2 −3x −4
g(x) = x2
−16
3x 2 +12x .
R(x) = f (x)
g(x)
g(x) = 9x2 +54x
x 2 − x −42 .
f (x) = 27x2
3x −21
7 + 8 15 15 4a 2
2a −1 − 2a 1
−1
y 2 +10y
y +5
+ 25
y +5
7x 2
x 2 −9 + 21x
x 2 −9
x 2 − 3x +28
x −7 x −7
y 2
y +11 − 121
y +11

4q 2 − q +3
q 2 +6q +5 − 3q2 − q −6
q 2 +6q +5
5t 2 +4t +3
t 2 − 4t2 −8t −32
−25 t 2 −25
18w + 3w −2
6w −1 1−6w a 2 +3a
a 2 − 3a +8
−4 4−a 2
2b 2 +3b −15
b 2 − b2 +16b −1
−49 49 − b 2
8y 2 −10y +7
+ 2y2 +7y +2
2y −5 5−2y
7
a 2 −3a −10 , 3a
a 2 − a −20
6
n 2 −4 , 2n
n 2 −4n +4
5
3p 2 +17p −6 , 2m
3p 2 +23p −8
7 + 3 2
5a 2b
c −2 + c +3
9
3x
x 2 −9 + 5
x 2 +6x +9
2x
x 2 +10x +24 + 3x
x 2 +8x +16
5q
p 2 q − p 2 + 4q
q 2 −1
3y
y +2 − y +2
y +8
−3w −15
w 2 + w −20 − w 4−w
+2
8a
a 2 −64 − 4
a +8
7m +3
m +2 −5 n
n +3 + 2
n −3 − n +9
n 2 −9
5
12x 2 y + 7
20xy 3
R(x) =f(x) +g(x) f (x) g(x)
f (x) = 2x2 +12x −11
x 2 , g(x) = x +1
+3x −10 2−x
f (x) =
−4x +31
x 2 + x −30 , g(x) = 5
x +6
R(x) =f(x) −g(x) f (x) g(x)
f (x) = 4x
x 2 − 121 , g(x) = 2
x −11
f (x) =
x +6 7 , g(x) = 14x
x 2 −36

7x
x +2
14x 2
x 2 −4
2
5 + 5 6
1
3 + 1 4
x − 3x
x +5
1
x +5 + 1
x −5
2
m + m n
n
m − 1 n
1
3 + 1 8
1
4 + 1 12
3
a 2 − 1 b
1
a + 1
b 2
2
z 2 + 1
−49
z +7
9
z +7 + z 12
−7
3
y 2 −4y −32
2
y −8 + 1
y +4
1
2
+ 2 3 = 1 x 1− 2 m = 8 m 2 1
b −2 + 1
b +2 = 3
b 2 −4
3
q +8 − 2
q −2 =1 v −15
v 2 −9v +18 = 4
v −3 + 2
v −6
z
12 + z +3 =
3z
1 z
f (x) = x +2
x 2 −6x +8 ,
b
a
f (x) =1 c
f (x) = 2−x
x 2 +7x +10 ,
b
a
f (x) =2 c
V = hw l. 1
l x − 2 y =5 y. x = y +5 z.
z −7
P = k V
V.

x = 3 3
4 5
y = 9 5
s
s +20 = 3 7
t −3
5
= t +2
9
ΔABC ΔXYZ.
a
b

y x
y =9 x =3,
y = 21.
x
y x
y =20 x =2,
x =4.
y
d,
v,
m
n, m =4
n =6 m n =2.
x −3
x +4 ≤0 5x
x −2 >1 3x −2
x −4 ≤2
1
x 2 −4x −12

R(x) = x x −5
−2 , R(x) = x x +1
+3 , C(x) = 150x + 100,000
x,
x x
a
c(x) b
C(x) =2x + 6,000
x,
a
c(x)
b

4a 2 b
12ab 2 6x −18
x 2 −9
4x
x +2· x2 +5x +6
12x 2
2y 2
y 2 −1 ÷ y3 − y 2 + y
y 3 +1
6x 2 − x +20
x 2 − 5x2 +11x −7
−81 x 2 −81
−3a
3a −3 + 5a
a 2 +3a −4
2n 2 +8n −1
n 2 − n2 −7n −1
−1 1−n 2
10x 2 +16x −7+ 8x −3
2x2 +3x −1
3−8x
1
m − 1 n
1
n + 1 m
1 x + 3 4 = 5 8
1
z −5 + 1
z +5 = 1
z 2 −25
z
2z +8 − 3
4z −8 = 3z2 −16z −16
8z 2 +16z −64
6x
x −6 ≤2 2x +3
x −6 >1 1 + 12
2 x 2 ≥ 5 x
R(x) f (x) = x−4
x 2 g(x) = x−5
−3x−10 x 2 −2x−8 .
R(x) = f (x)−g(x) R(x) = f (x)·g(x) R(x) = f (x) ÷g(x)
R(x) = 2
2x 2 ,
+ x −15
x
y x
x =5 y = 30, x
y = 42.
y
x x =3
y =9, y x =4.

a (−9) 2 b −9 2 c (−9) 3 .

3.846
a x 3 · x 3 b y 2 · y 2 · y 2 c z 3 · z 3 · z 3 · z 3 .
n 2
13 2 is read “13 squared”
169 is called the square of 13, since 13 2 = 169
13 is a square root of 169
If n 2 = m, then m is the square of n.
If n 2 = m, then n is a square root of m.
m,
0 2 =0,
0 =0.
m is read “the square root of m”.
If n 2 = m, then n = m, forn ≥0.
169 = 13.
− 169 = −13.
a 144 b − 289.
a

144
12 2 = 144. 12
b
− 289
17 2 = 289
−17
a − 64 b 225.
a 100 b − 121.
−49? −49?
() 2 = −49
−49.
a −196 b −64.
a
−196
−196.
−196 is not a real number.
b
− 64
−8
a −169 b − 81.
a − 49 b −121.

We write:
n 2
n 3
n 4
n 5
We say:
n squared
n cubed
n to the fourth power
n to the fifth power
If b n = a, then b is an n th root of a.
The principal n th root of a is written n a.
n is called the index of the radical.
4 3 = 64
3 4 = 81
(−2) 5 = −32
3 a.
3
64
4
81
5
−32
= 4
= 3
= −2
n
a
a ≥0, n a
a

3
a 64
a
4
b 81
5
c 32.
3
64
4 3 = 64. 4
b
4
81
(3) 4 = 81. 3
c
5
32
(2) 5 = 32. 2
3
a 27
4
b 256
5
c 243.
3
a 1000
4
b 16
5
c 1024.
3
a −125
a
4
b 16
5
c −243.
3
−125
(−5) 3 = −125.
−5
b

4
−16
(?) 4 = −16.
c
5
−243
(−3) 5 = −243.
−3
3
a −27
4
b −256
5
c −32.
3
a −216
4
b −81
5
c −1024.
3
4 =2, 21 50?
11,
3
91,
3
a 105 b 43.
a

105
b
3
43
3
a 38 b 93
3
a 84 b 152
x
y
x
≈
4
93
5 ≈ 2.236067978 rounded to two decimal places is 5 ≈ 2.24
4
≈ 3.105422799 rounded to two decimal places is 93
≈ 3.11
(2.236067978) 2 = 5.000000002
(2.24) 2 = 5.0176
(3.105422799) 4 = 92.999999991
(3.11) 4 = 93.54951841

3
a 17 b 49
a
4
c 51.
17
4.123105626…
4.12
17 ≈ 4.12
b
3
49
y
x
3.659305710…
3.66
3
49
≈ 3.66
c
4
51
y
x
2.6723451177…
2.67
4
51
≈ 2.67
3
a 11 b 71
4
c 127.
3
a 13 b 84
4
c 98.

4
625 =5.
|−5| =5.
n
a n = |a|.
n ≥2,
n
when the index n is odd
a n = a
n
when the index n is even
a n = |a|
a x 2 3
b n 3
4
c p 4
5
d y 5 .
a
n
n
a n
x 2
= |a|. |x|
b
n
n
a n
3
m 3
= a. m
c

n
n is even a n
4
p 4
= |a|. |p|
d
n
n
a n
5
y 5
= a. y
a b 2 3
b w 3
4
c m 4
5
d q 5 .
a y 2 3
b p 3
4
c z 4
5
d q 5 .
(a m ) n = a m · n .
(a m ) 2 = a 2m
a 2m
Since (a m ) 2 = a 2m . (a m ) 2
n
Since n is even a n = |a|. |a m |
So a 2m = |a m |.
a x 6 b y 16 .
a
x 6
⎛ ⎝x 3⎞ ⎠2
= x 6 ⎛
. ⎝ x 3⎞ 2
⎠
n a n = |a|. | x3 |
b

y 16
⎛ ⎝y 8⎞ ⎠2
= y 16 .
⎛
⎝ y 8⎞ ⎠
n
n
a n
= |a|. y 8
y 8
2
a y 18 b z 12 .
a m 4 b b 10 .
3
a y 18
4
b z 8 .
a
⎛ ⎝y 6⎞ ⎠
n
3
y 18
3
= y 18 3
⎛
. ⎝ y 6⎞ ⎠3
n
a n
= a. y 6
b
⎛ ⎝z 2⎞ ⎠
4
z 8
4
= z 8 4
⎛
. ⎝ z 2⎞ ⎠4
z 2 z 2
4
a u 12
3
b v 15 .
5
a c 20
6
b d 24
a 2m = |a m |

16r 22 =4 |
r 11 | because ⎛ ⎝4r 11⎞ ⎠2
=16r 22 .
25u 8 =5u 4
a 16n 2 b −81c 2 .
a
16n 2
(4n) 2 =16n 2 . (4n) 2
n
n
a n
= |a|. 4|n|
b
− 81c 2
(9c) 2 =81c 2 . − (9c) 2
n
n
a n
= |a|. −9|c|
a 64x 2 b − 100p 2 .
a 169y 2 b − 121y 2 .
3
a 64p 6
4
b 16q 12 .
a
3
64p 6
64p 6 ⎛ ⎝4p 2⎞ 3 3
⎠ .
⎛
⎝ 4p 2⎞ ⎠3
4p 2

4
16q 12
4
⎛
⎝ 2q 3⎞ 4
⎠
2 |
q 3 |
3
a 27x 27
4
b 81q 28 .
3
a 125q 9
5
b 243q 25 .
a 36x 2 y 2 b 121a 6 b 8 3
c 64p 63 q 9 .
a
36x 2 y 2
⎛ ⎝6xy ⎞ ⎠2 =36x 2 y 2 ⎛ ⎝6xy ⎞ 2
⎠
6|xy|
b
121a 6 b 8
⎛ ⎝11a 3 b 4⎞ ⎠2
= 121a 6 b 8 ⎛ ⎝11a 3 b 4⎞ 2
⎠
11 |
a 3 | b4
c

⎛ ⎝4p 21 q 3⎞ ⎠
3
64p 63 q 9
3
=64p 63 q 9 3
⎛ ⎝4p 21 q 3⎞ ⎠3
4p 21 q 3
a 100a 2 b 2 b 144p 12 q 20 3
c 8x 30 y 12
a 225m 2 n 2 b 169x 10 y 14 3
c 27w 36 z 15

a 64 b −81 a 169 b − 100 a 196 b −1
a 144 b − 121 a 4 9
b − 0.01 a 64
121 b − 0.16
a −121 b − 289 a − 400 b −36 a − 225 b −9
3
a −49 b − 256 a 216
4
b 256
3
a 27
4
b 16
5
c 243
3
a 512
4
b 81
5
c 1
3
a 125
4
b 1296
5
c 1024
3
a −8
4
b −81
5
c −32
3
a −64
4
b −16
5
c −243
3
a −125
5
−1024
b
4
−1296
c
3
a −512
4
b −81
5
c −1
3
a 70 b 71
3
a 172 b 200
3
a 55 b 119
3
a 200 b 137
3
a 19 b 89
4
c 97
3
a 21 b 93
4
c 101
3
a 53 b 147
4
c 452
3
a 47 b 163
4
c 527
5
a u 5
8
b v 8
3
a a 3
9
b b 9
4
a y 4
7
b m 7
8
a k 8
6
b p 6
a x 6 b y 16 a a 14 b w 24
a x 24 b y 22 a a 12 b b 26 3
a x 9
4
b y 12
5
a a 10
3
b b 27
4
a m 8
5
b n 20
6
a r 12
3
b s 30
a 49x 2 b −81x 18 a 100y 2 b − 100m 32 a 121m 20 b −64a 2

a 81x 36 b −25x 2 4
a 16x 8 6
b 64y 12
3
a −8c 9
3
b 125d 15
3
a 216a 6
5
b 32b 20
7
a 128r 14
4
b 81s 24
a 144x 2 y 2 b 169w 8 y 10
3
c 8a 51 b 6
a 196a 2 b 2 b 81p 24 q 6
3
c 27p 45 q 9
a 121a 2 b 2 b 9c 8 d 12 c
3
64x 15 y 66
a 225x 2 y 2 z 2 b 36r 6 s 20
3
c 125y 18 z 27
−64? 9 2 9?
a
b

x 9
x 4.
y 3
y 11.
⎛ ⎝ ⎛ n 2⎞ ⎠ ⎞6 .
n a,
m n .
n ≥2,
n
a is considered simplified if a has no factors of m n
5 12
3
3
4 24
(ab) n = a n b n .
n
ab
= n a · n b.
n n
a b n ≥2
n
ab = n a · n b and n a · n n
b = ab

98.
48.
45.
98 72,
7
72 2 .
3
a 500 b 16
a
4
c 243.

500
100 · 5
100 · 5
10 5
b
2 3 3
=8 8·2
3 3
8 · 2
2 3 2
3
16
c
4
243
3 4 =81
4
81 · 3
4
81 · 4 3
3 4 3
3
a 288 b 81
4
c 64.
3
a 432 b 625
4
c 729.
a x 3 3
b x 4
a
4
c x 7 .

x 3
x 2 · x
x 2 · x
|x| x
b
3
x 4
3
x 3 · x.
3
x 3 3
· x
3
x x
c
4
x 4 · x 3
4
x 4 4
· x 3
4
|x| x 3
4
x 7
a b 5 4
b y 6
3
c z 5
a p 9 5
b y 8
6
c q 13
a 72n 7 3
b 24x 7
4
c 80y 14 .
a

72n 7
36n 6 ·2n
36n 6 · 2n
6 |
n 3 |
2n
b
3
24x 7
3
8x 6 ·3x
3
8x 6 3
· 3x
⎛ ⎝2x 2⎞ ⎠
3 3
.
⎛
⎝ 2 x 2⎞ ⎠3
2x 2 3
3 x
3
· 3x
c
4
80y 14
4
16y 12 ·5y 2
4
16y 12
4
· 5y 2
⎛ ⎝2y 3⎞ ⎠
4 4
. ⎛
⎝ 2 y 3⎞ ⎠4
2 |
y 3 4
|
5 y 2
4
· 5y 2
a 32y 5 3
b 54p 10
4
c 64q 10 .
a 75a 9 3
b 128m 11
4
c 162n 7 .

a 63u 3 v 5 3
b 40x 4 y 5
4
c 48x 4 y 7 .
a
63u 3 v 5
9u 2 v 4 ·7uv
9u 2 v 4 · 7uv
⎛ ⎝3uv 2⎞ ⎠2
.
⎛
⎝ 3uv 2⎞ ⎠2
· 7uv
3|u|v 2 7uv
b
3
40x 4 y 5
3
8x 3 y 3 ·5xy 2
3
8x 3 y 3 3
· 5xy 2
⎛ ⎝2xy ⎞ ⎠3 3 3
.
⎛
⎝2xy ⎞ ⎠3
· 5xy 2
3
2xy 5xy 2
c
4
48x 4 y 7
4
16x 4 y 4 ·3y 3
4
16x 4 y 4 4
· 3y 3
⎛ ⎝2xy ⎞ ⎠4 4 4
.
⎛
⎝2xy ⎞ ⎠4
· 3y 3
4
2|xy| 3y 3

a 98a 7 b 5 3
b 56x 5 y 4
4
c 32x 5 y 8 .
a 180m 9 n 11 3
b 72x 6 y 5
4
c 80x 7 y 4 .
3
a −27
a
4
b −16.
3
−27
3
(−3) 3
−3
b
4
−16
n n 4 = −16.
3
a −64
4
b −81.
3
a −625
4
b −324.
a 3+ 32 b
a
4− 48.
2

3+ 32
3+ 16·2
3+ 16· 2
3+4 2
b
4− 48
2
4− 16·3
2
4− 16· 3
2
4−4 3
2
4 ⎛ ⎝1− 3 ⎞ ⎠
2
2 ·2 ⎛ ⎝1− 3 ⎞ ⎠
2
2 ⎛ ⎝1− 3 ⎞ ⎠
a 5 + 75 b
10 − 75
5
a 2 + 98 b 6 − 45
3
a 45
80 b 3
16
54
a
4
c
80
5 .

45
80
⎛ ⎝ 3 ⎞
4⎠2 = 9 . 16
5·9
5·16
9
16
3
4
b
3
16
54
3
2·8
2·27
3
8
27
⎛ ⎝ 2 3
⎞
3⎠
=
27 8 . 2
3
c
4 5
80
⎛ ⎝ 1 ⎞
2⎠4 = 1 . 16
4 5·1
5·16
4 1
16
1
2
a 75
48 b 3
250
54
4
c
162
32 .

a
162 98 b 3
375
24
4
c
324
4 .
a m
a n = am − n , a ≠0
a
a
m6 b
m 4
3
a8
a 5
c
4
a10
a 2 .
m 6
m 4
m 2
|m|
b
3
a 3
3
a 8
a 5
a
c
4
a 8
4
a 10
a 2
4
⎛
⎝ a 2⎞ 4
⎠
a 2

a
a8 b
a 6
4
x7
x 3
c
4
y17
y 5 .
a
x14 b
x 10
3
m13
m 7
c
5
n12
n 2 .
⎛
⎝ a m
⎞
b⎠
= am
b
m, b ≠0
n n
a b b ≠0, n ≥2
n a
b
= n a
b
n and n a n
n = a
b b
27m 3
196 .
24p 3
49 .
48x 5
100 .

a
45x5 3
b 24x7
y 4 y 3
c
4
48x10
y 8 .
a
45x 5
y 4
45x 5
y 4
9x 4 · 5x
y 2
3x 2 5x
y 2
b
3
24x 7
y 3
3
24x 7
3
y 3
3
8x 6 ·3x
3
y 3
3
⎛
⎝2x 2⎞ 3
⎠
3
y 3
3
· 3x
2x 2 3
3x
y

c
4
48x 10
y 8
4
48x 10
4
y 8
4
16x 8 ·3x 2
4
y 8
4
⎛
⎝2x 2⎞ 4
⎠
4
⎛
⎝y 2⎞ 4
⎠
4
· 3x 2
2x 2 4
3x 2
y 2
a
80m3 3
b 108c10
n 6 d 6
c
4
80x10
y 4 .
a
54u7 3
b 40r3
v 8 s 6
c
4
162m14
n 12 .
a 18p5 q 7
b 3
16x5 y 7
32pq 2 54x 2 y 2
4
c 5a8 b 6
80a 3 b 2 .
a

18p 5 q 7
32pq 2
9p 4 q 5
16
9p 4 q 5
16
9p 4 q 4 · q
4
3p 2 q 2
4
q
b
3
16x 5 y 7
54x 2 y 2
3
8x 3 y 5
27
3
8x 3 y 5
3
27
3
8x 3 y 3
3
27
3
· y 2
3
2xy y 2
3
c

4
5a 8 b 6
80a 3 b 2
4
a 5 b 4
16
4
a 5 b 4
4
16
4
a 4 b 4 · 4 a
4
16
|ab| 4 a
2
a 50x5 y 3
b 3
16x5 y 7
72x 4 y 54x 2 y 2
4
c 5a8 b 6
80a 3 b 2 .
a 48m7 n 2
b 3
54x7 y 5
100m 5 n 8 250x 2 y 2
4
c 32a9 b 7
162a 3 b 3 .
a
a
48a7
3a b 3
−108
4
c 96x7 .
3 4
2 3x 2
48a 7
3a
16a 6
4 |
a 3 |
48a 7
3a
b

3
−108
3
2
3
−108
2
3
−54
3
(−3) 3 ·2
3
(−3) 3 3
· 2
3
−3 2
c
4
96x 7
4
3x 2
4
96x 7
3x 2
4
32x 5
4
16x 4 4
· 2x
4
(2x) 4 4
· 2x
4
2|x| 2x
a
98z5
2z
b
3
−500
4
c 486m11 .
3 4
2 3m 5
a
128m9
2m b 3
−192
4
c 324n7 .
3 4
3 2n 3

27 80 125
96 147 450
800 675 4
a 32
5
b 64
3
a 625
6
b 128
4
a 64
3
b 256
4
a 3125
3
b 81
a y 11 3
b r 5
4
c s 10
a m 13 5
b u 7
6
c v 11
a n 21 3
b q 8
8
c n 10
a r 25 5
b p 8
4
c m 5
a 125r 13 3
b 108x 5
c
a 80s 15 5
b 96a 7
c
4
48y 6
6
128b 7
a 242m 23 4
b 405m 10
c
a 175n 13 5
b 512p 5
c
a 147m 7 n 11 3
b 48x 6 y 7
5
160n 8
4
324q 7
4
c 32x 5 y 4
a 96r 3 s 3 3
b 80x 7 y 6
c
a 192q 3 r 7 3
b 54m 9 n 10
a 150m 9 n 3 3
b 81p 7 q 8
4
80x 8 y 9
4
c 81a 9 b 8
4
c 162c 11 d 12
3
a −864
4
b −256
5
a −486
6
b −64
5
a −32
8
b −1
3
a −8
4
b −16
a 5+ 12 b
10 − 24
2
a 8+ 96 b
8− 80
4
a 1+ 45 b
3+ 90
3
a 3 + 125 b
15 + 75
5
a 45
80 b 3
27
8
4
c
81
1
a 72
98 b 3
24
81
4
c
96
6
a 100
36 b 3
375
81
4
c
256
1
a 121
16 b 3
250
16
4
c
162
32
a
x10 b 3
p11
x 6 p 2
c
4
q17
q 13
a
p20
p 10 b
5
d12
d 7
c
8
m12
m 4

a
y4
y 8 b
5
u21
u 11
c
6
v30
v 12
a
q8
q 14 b
3
r14
r 5
c
4
c21
c 9
96x 7
121
108y 4
49
300m 5
64
125n 7
169
98r 5
100
180s 10
144
28q 6
225
150r 3
256
a 75r9 3
b 54a8
s 8 b 3
4
64c 5
d 4
c
a
6
128u 7
v 12
72x5 5
b 96r11
y 6 s 5
c
a 28p7 3
b 81s8
q 2 t 3
4
64p 15
q 12
c
a
4
729c 21
d 8
45r3 3
b 625u10
s 10 v 3
c
a 32x5 y 3
b 3
5x6 y 9
18x 3 y 40x 5 y 3
4
5a 8 b 6
80a 3 b 2
c
a 75r6 s 8 3
b 24x8 y 4
48rs 4 81x 2 y
4
32m 9 n 2
162mn 2
c
a
6
c 2m9 n 7
128m 3 n
27p2 q
b 3
16c5 d 7
108p 4 q 3 250c 2 d 2
a 50r5 s 2
b 24m9 n 7
128r 2 s 6 375m 4 n
4
c 81m2 n 8
256m 1 n 2
3
a 45p9 4
b 64 c
5q 2 4
2
5
128x 8
5
2x 2
a
4
80m 7
4
5m
80q5
5q
b
3
−625 c
3
5
a
4
486y 9
2y 3
50m7
2m b 3
1250
2
c
a 72n11
2n
4
160r 10
5r 3
3
b 162
6
c
x 4 = x 2 .
x 16 = x 8 .
7+ 9 7+9.
5
x 10
= x 2 4
. −64
3
−64

a
b

1
n
a
m
a n
7
15
+ 5 12 .
⎛ ⎝ ⎛ 4x 2 y 5⎞ ⎠ ⎞3 .
5 −3 .
1
n
a
(a m ) n = a m · n
(8 p ) 3 =8.
(8 p ) 3 = 8
Multiply the exponents on the left. 8 3p = 8
Write the exponent 1 on the right. 8 3p = 8 1
Since the bases are the same, the exponents must be equal. 3p = 1
Solve for p. p = 1 3
⎛ 1 ⎞ 3
3 ⎜8
⎝ ⎠ ⎟
=8. ⎛ 3
3
⎝ 8 ⎞
⎠
1
=8. 8
3 = 3 8.
1
a
n =
n a.
1
n
a
n a
n ≥2,

1
a
n =
n a
1 1 1
a x 2
b y 3
c z
4 .
a
n a.
1
2
x
x
b
1
3
y
3
y
c
1
4
z
4
z
1 1 1
a t 2
b m 3
c r
4 .
1 1 1
a b 6
b z 5
c p
4 .
3
a 5y b 4x
4
c 35z.

1
a
n .
a
5y
1
⎛
⎝5y ⎞ 2
⎠
5y.
b
3
4x
(4x).
1
3
(4x)
c
4
3 5z
5z
1
4
3(5z)
5
a 10m b 3n
4
c 3 6y.
7
a 3k
4
b 5j
3
c 8 2a.
1 1 1
a 25 2
b 64 3
c 256
4 .
a

1
2
25
25
5
b
1
3
64
3
64
3
4 3
4
c
1
4
256
4
256
4
4 4
4
1 1 1
a 36 2
b 8 3
c 16
4 .
1 1 1
a 100 2
b 27 3
c 81
4 .
a −n = 1 n
a
1 1
a (−16) 4
b −16 4
c (16) − 14 .
a

1
4
(−16)
4
−16
4
(−2) 4
No real solution.
b
1
4
−16
4
−16
2 4 4
. −2 4
−2
c
(16) − 1 4
a −n = 1 1
a n. 1
4
(16)
2 4 .
1
4
16
1
4
2 4
1
2
a (−64) − 1 1
2
b −64 2
c (64) − 12 .
1
1
a (−256) 4
b −256 4
c (256) − 14 .
m
a n
m
a n
⎛ 1
n ⎜a
⎞ m
⎝ ⎠ ⎟

1
(a m n
)
a
m n
.
m
a n
= ⎛ ⎜a 1
m
⎞ n ⎟
⎝ ⎠
= ( n a) m m
and a n
= (a m 1
) n =
n
a m
m
a n
m
a n
= ( n a) m m
and a n n
= a m
a
m
a n n
= a m
a
y 3 b ⎛ 4
3
3
⎝ 2x ⎞
⎠
c
⎛3a⎞ ⎝4b⎠ .
a
m n
.
b
c

a
x 5 b ⎛ ⎝ 4 ⎞ 3 3y⎠
c ⎛2m⎞ 5
⎝ 3n ⎠ .
5
a a 2
b ⎛ ⎝ 3 ⎞ 5 5ab⎠
c ⎛7xy
⎛ x ⎞ 3 ⎝ z ⎠
.
a −n = 1 a
n.
2
a 125 3
b 16 − 3 2
c 32 − 25 .
m
a n
= ( n a) m .
a
2
3
125
2
⎛3 ⎝ 125 ⎞
⎠
(5) 2
25
b a −n = 1 a n
16 − 3 2
a −n = 1 1
a n 3
2
16
1
⎛
⎝ 16 ⎞3
⎠
1
4 3
1
64
c

32 − 2 5
a −n = 1 1
a n. 2
5
32
1
2
⎛5 ⎝ 32 ⎞
⎠
1
2
⎛5 ⎞
⎝ 25 ⎠
1
2 2
1
4
2
a 27 3
b 81 − 3 2
c 16 − 34 .
3
a 4 2
b 27 − 2 3
c 625 − 34 .
3
a −25
2
b −25 − 3 2
c (−25)
3
2 .
a
3
2
−25
− ⎛ ⎝ 25 ⎞ ⎠ 3
−(5) 3
−125
b

−25 − 3 2
a −n = 1 a n.
⎛ ⎞
− ⎜ 1 ⎟
⎜ 3 ⎟
⎝ 2
25 ⎠
⎛ ⎞
−⎜ 1
⎛
⎝ 25 ⎞3 ⎟
⎠ ⎠
⎝
− ⎛ ⎝ ⎜ 1 ⎞
(5) 3 ⎟
⎠
− 1
125
c
3
2
(−25)
is −25.
⎛
⎝ −25 ⎞3
⎠
Not a real number.
3
a −16 2
b −16 − 3 2
c (−16) − 32 .
3
a −81 2
b −81 − 3 2
c (−81) − 32 .
Product Property a m · a n = a m + n
Power Property (a m ) n = a m · n
Product to a Power (ab) m = a m b m
Quotient Property
a m
a n = a m − n , a ≠0
Zero Exponent Definition a 0 = 1, a ≠0
Quotient to a Power Property
⎛
⎝ a m
⎞
b⎠
= am
b
m, b ≠0
Negative Exponent Property a −n = 1 a
n, a ≠0

2
1 5
1
a x 2 · x 6
b ⎛ ⎝z 9⎞ 3 3
⎠ c x .
5
3
x
a
1 5
x 2 6
· x
1
2
x
+ 5 6
8
6
x
4
3
x
b
2
⎛
⎝z 9⎞ 3
⎠
9·23
z
z 6
c
1
3
x
5
3
x
1
3
x
5
3
x
1
5
3
x
− 1 3
1
4
3
x

4
1
a x 6 4
2
· x 3
b 3
⎛ 6⎞⎞
3
⎝x
⎠ c x .
5
3
x
2
3 5
1
a y 4 · y 8
b ⎛ ⎝ m 9⎞ ⎠ ⎞ 9 5
c d .
6
5
d
a ⎛ 1
⎝ ⎜27u ⎞
2
⎟
⎠
2
3
3
b ⎛ 2 1 ⎞ 2
⎝ ⎜m 3 2
n
⎠ ⎟ .
a
2
⎛ 1
2 ⎜27u
⎞ 3
⎝ ⎠ ⎟
2
2 ⎛ 1
3
3
(27)
⎝ ⎜u ⎞
2
⎟
⎠
2
⎛
⎝ 3 3⎞ 3 ⎛
⎠
⎛
⎝3 2⎞ ⎠
2
1
2 ⎜u
⎞ 3
⎝ ⎠ ⎟
⎛ 1 ⎞ 3 ⎜u
⎝ ⎠ ⎟
1
3
9u
b

⎛
⎜m
⎝
2
3 1
n
3
⎞ 2
2
⎠ ⎟
3
2
3
⎛ 2 ⎞ 2 ⎛ 1
3 ⎜m
⎝ ⎠ ⎟ ⎝ ⎜n ⎞
2
⎟
⎠
3
4
mn
a ⎛ 1 ⎜⎛
⎞
⎝ 32x 3
⎟
⎠
3
5
b ⎛ ⎝ ⎜⎛
⎝ x 3
4 1
y
2 ⎞ ⎠ ⎟⎞
⎠
2
3
.
a ⎛ 2 ⎜⎛
⎞
⎝ 81n 5
⎟
⎠
3
2
b ⎛ ⎝ ⎜⎛
⎝ a 3
2 1
b
2 ⎞ ⎠ ⎟⎞
⎠
4
3
.
a x 3
4 · x − 1 4
x − 6 4
⎛
b ⎜
⎝
1
4
⎜16 x
3 y − 5 2
⎞
6⎟
x − 23 1 ⎟
.
6
y ⎠
a
3
x 4 · x − 1 4
x − 6 4
2
4
x
x − 6 4
8
4
x
x 2
b

⎛ 4
⎜16 x
⎜
⎝ x − 23 y
3 y − 5 6
1
6
1
2
⎞
⎟
⎟
⎠
⎛ 6⎞
⎜ 16x
3⎟
⎜ ⎟
⎝ ⎠
6
6
y
1
2
⎛
⎝
16x 2
y
⎞ ⎠
1
2
4x
1
2
y
a m 23
· m − 1 3
m − 5 3
b
1
⎛ 1 11
2
⎞
⎜
⎜ 25m 6
n 6 ⎟
2
⎝ m
3 n − 1 ⎟
.
6
⎠
a u 45
· u − 2 5
u − 13 5
b
1
⎛ 4 1
3
⎞
⎜
⎜ 27x 5 6
y ⎟
1
⎝ x
5 y − 5 ⎟
.
6
⎠

1
n
a
1 1 1
a x 2
b y 3 4
c z
1 1 1
a r 2
b s 3 4
c t
1 1 1
a u 5
b v 9 20
c w
1 1 1
a g 7
b h 5 25
c j
a
7 9 5
x b y c f
8
a r b 10 4
s c t
3
a 7c
7
b 12d
4
c 26b
4
a 5x
8
b 9y
5
c 73z
4
a 21p b 8q
6
c 436r
3
a 25a
8
b 3b c 40c
1 1 1
a 81 2
b 125 3 2
c 64
1 1 1
a 625 4
b 243 5 5
c 32
1 1 1
a 16 4
b 16 2 4
c 625
1 1 1
a 64 3
b 32 5 4
c 81
1
1
a (−216) 3
b −216 3
c
(216) − 1 3
1
1
a (−1000) 3
b −1000 3
c
(1000) − 1 3
1
1
4 4
a (−81) b −81 c
1
1
2 2
a (−49) b −49 c
1
1
2 2
a (−36) b −36 c
(81) − 1 4
1
1
4 4
a (−16) b −16 c
16 − 1 4
(49) − 1 2
1
1
a (−100) 2
b −100 2
c
(100) − 1 2
(36) − 1 2
1
a (−32) 5
b (243) − 1 5
c
1
3
−125
m
a n
a m 5 b ⎛ 7
3
⎝ 3y ⎞
⎠
c
4
a r 7
b
⎛ 3
5
⎝ 2pq ⎞
⎠
c
5
a u 2
b
⎛ 5
3
⎝ 6x ⎞
⎠
c
5
⎛
⎝ 4x ⎞
5y⎠
3
4
⎛
⎝ 12m 3
⎞
7n ⎠
4
⎛
⎝ 18a 7
⎞
5b ⎠

a 3 a b ⎛ 3
4
⎝ 21v ⎞
⎠
c
4
⎛
⎝ 2xy 2
⎞
5z ⎠
2
2 3
c (−27)
5 −3
a 64 2
b 81
3
a 25
2
b 9 − 3 2
c (−64)
2
3
2
a 32
5
b 27 − 2 3
c (−25)
1
2
3
a 100
3
2
(−100)
2
b 49 − 5 2
c
3
a −9
2
b −9 − 3 2
c (−9)
3
2
3
a −64
3
2
(−64)
2
b −64 − 3 2
c
3
1 5
4
3
a c 4 · c 8
b ⎛ ⎝p 12⎞ 4 5
5
⎠ c r a 6 2 1
2
·6 2
b ⎛ 9
⎝b 15⎞ 5 7
⎠ c w
9
5
7
r
w 5
8
m
m
13
8
1
a y
3
2 4
· y b ⎛ ⎝x 12⎞ ⎠
2
3
c
4
2 5
3
a q 3 · q 6
b ⎛ ⎝h 6⎞ 3 5
⎠ c n
8
5
n
4 3
a ⎛ 3 ⎞ 3
⎝ ⎜27q 2
⎠ ⎟ b ⎛ 1 2 ⎞ 2
⎝ ⎜a 3 3
b
⎠ ⎟
a ⎛ 3⎞
⎝ ⎜64s 7
⎟
⎠
1
6
3
b ⎛ 4 1 ⎞ 4
⎝ ⎜m 3 2
n
⎠ ⎟
a ⎛ 1⎞
⎝ ⎜16 u 3
⎟
⎠
3
4
3
b ⎛ 1 1 ⎞ 2
⎝ ⎜4 p 3 2
q
⎠ ⎟
3
a ⎛ 8 ⎞ 4
⎝ ⎜625 n 3
⎠ ⎟
5
b ⎛ 2 3 ⎞ 2
⎝ ⎜9 x 5 5
y
⎠ ⎟
5
a r 2 · r − 1 2
1
⎛ 1
⎜36 s
5 t − 3 2
⎞
2 ⎟
⎜
⎝ s − 95 1 ⎟
2
t ⎠
r − 3 2
b
3
a a 4 · a − 1 4
1
⎛ 2
⎜27 b
3 c − 5 3
⎞
2 ⎟
⎜
⎝ b − 73 1 ⎟
2
c ⎠
a − 10 4
b
5
a c 3 · c − 1 3
⎛ 5
⎜ 8 x
⎜
⎝27 x − 43 y
3 y − 1 2
5
2
⎞
⎟
⎟
⎠
1
3
c − 2 3
b
7
a m 4 · m − 5 4
1
⎛ 1
4
⎜ 16 m
5 3 ⎞
2
n ⎟
⎜ 9
⎝81 m
5 n − 1 ⎟
2 ⎠
m − 2 4
b
3
4
2 .
3
2
(−16)

a
b

3x 2 + 9x − 5 − ⎛
⎝x
2 − 2x + 3 ⎞ ⎠ .
(2 + a)(4 − a).
⎛ ⎝ ⎛ 9 − 5y ⎞ ⎠ ⎞2 .
3x +8x 11x.
3 x +8 x 11 x.
a 2 2−7 2 b 5 3 3 4
y +4 y c 7 x −2 4 y .
a
2 2−7 2
−5 2
b

5 3 3
y +4 y
3
9 y
c
4
7 x −2
4 y
3
a 8 2 − 9 2 b 4 x +7
3 x
4
c 3 x − 5 4 y .
a 5 3 − 9 3 b 5 3 3
y + 3 y c 5 4 m − 2 3 m.
4
a 2 5n −6 5n +4 5n b 3xy
4
+5 3xy
4
−4 3xy.
a
2 5n −6 5n +4 5n
0 5n
0
b
4
3xy
4 4
+5 3xy −4 3xy
4
2 3xy
4
a 7x − 7 7x + 4 7x b 4 5xyx
4
+ 2 5xyx
4
− 7 5xyx
.
3
a 4 3y − 7 3y + 2 3y b 6 7mn
3
+ 7mn
3
− 4 7mn.

3 3
a 20 +3 5 b 24 − 375
c 1 4
2 48
− 2 4
3 243 .
a
20 +3 5
4 · 5+3 5
2 5+3 5
5 5
b
3 3
8 · 3
3 3
24 − 375
3
− 125 · 3 3
2 3 3
3−5 3
3
−3 3
c
1 4
2 48
− 2 4
3 243
1 4
2 16 · 4 3− 2 4
3 81 · 4 3
1
2 ·2· 4
3
− 2 3 ·3· 4
3
4 4
3 −2 3
−3
4
3 3
a 18 + 6 2 b 6 16 − 2 250
c 2 3
3 81
− 1 3
2 24 .
3 3
a 27 + 4 3 b 4 5 − 7 40
c 1 3
2 128
− 5 3
3 54 .

a 950m 2 −6 48m 2 3
b 54n 5
a
3
− 16n 5 .
9 50m 2 −6 48m 2
9 25m 2 · 2−6 16m 2 · 3
9·5m · 2−6·4m · 3
45m 2 −24m 3
b
3
27n 3
3
54n 5 3
− 16n 5
3
· 2n 2
3
− 8n 3 3
· 2n 2
3
3n 2n 2 3
−2n 2n 2
3
n 2n 2
a 32m 7 − 50m 7 3
b 135x 7
3
− 40x 7 .
a 27p 3 − 48p 3 3
b 256y 5
3
− 32n 5 .
n n
a b , n ≥2
n
ab
= n a · b
n and n a
· n n
b = ab
4x ·3y

a ⎛ ⎝6 2 ⎞ ⎠ ⎛ ⎝3 10 ⎞ ⎠ b ⎛ ⎝ −5 3 4 ⎞⎛ 3 ⎞
⎠⎝ −4 6 ⎠ .
a
⎛
⎝6 2 ⎞ ⎠ ⎛ ⎝3 10 ⎞ ⎠
18 20
18 4 · 5
18·2· 5
36 5
b
⎛
⎝ −5 3 4 ⎞⎛ ⎠⎝ −4 3 ⎞
6 ⎠
3
20 24
3 3
20 8 · 3
20 · 2 · 3 3
3
40 3
a ⎛ ⎝ 3 2 ⎞⎛ ⎠⎝ 230 ⎞ ⎠⎞ b ⎛ ⎝ 3 ⎞⎛ 3 ⎞ 2 18⎠
⎝−3 6⎠.
a ⎛ ⎝ 3 3 ⎞ ⎞⎛ ⎠⎝ 3 6 ⎞ ⎠⎞ b ⎛ ⎝ 3 ⎞⎛ 3 ⎞ −4 9⎠
⎝3 6⎠.
a ⎛ ⎝ 10 6p3 ⎞⎛ ⎠⎝ 4 3p ⎞ ⎠ b ⎛ ⎝ 2 4 ⎞⎛ 20y2 4
⎠⎝ 3 28y 3 ⎞
⎠
.
a

⎛
⎝ 10 6p3 ⎞
⎠
⎛
⎝ 43p ⎞ ⎠
40 18p 4
40 9p 4 · 2
40 · 3p 2 · 3
120p 2 3
b
⎛
⎝ 220y2 4 ⎞⎛ 4
⎠⎝ 328y 3 ⎞
⎠
4
6 4·5·4·7y 5
4
6 16y 4 4
· 35y
4
6·2y 35y
4
12y 35y
a ⎛ ⎝ 6 6x
2⎞⎛ ⎠⎝8 30x
4⎞ ⎞ ⎠ b ⎛ ⎝ 4 −4 12y
3
⎞⎛ 4
⎠⎝ − 8y
3⎞ ⎠ .
a ⎛ ⎝ 2 6y
4⎞⎞ ⎛⎝ ⎠ 12 30y ⎞ ⎠⎞ b ⎛ ⎝ 4
−4 9a
3⎞⎛ 4
⎠⎝ 3 27a
2⎞ ⎠ .
a 6 ⎛ ⎝ 2 + 18 ⎞ 3 ⎛ 3 ⎞
⎠ b 9 ⎝ 5− 18 ⎠ .
a

6 ⎛ ⎝ 2 + 18 ⎞ ⎠
12 + 108
4 · 3+ 36· 3
23+6 3
83
b
3 ⎛ 3 ⎞
9 ⎝ 5− 18 ⎠
3 3
59− 162
3
5 9
3 3
− 27· 6
3 3
5 9−3 6
a 6 ⎛ ⎝ 1 + 3 6 ⎞ 3 ⎛ 3 ⎞ ⎠⎞ b 4 ⎝−2 − 6⎠.
a 8 ⎛ ⎝ ⎛ 2 − 5 8 ⎞ ⎠⎞ b 3
3 ⎛ 3
⎝− 9
3 ⎞ − 6 ⎠.
a (3−2 7)(4−2 7) b ⎛ 3
⎝ x −2
⎞⎛3 ⎠⎝ x +4
⎞
⎠ .
a
(3−2 7)(4−2 7)
12 − 6 7 −8 7+4·7
12 − 6 7 −8 7+28
40 − 14 7
b

⎛3 ⎝ x −2
⎞⎛3 ⎠⎝ x +4
⎞
⎠
3
x 2 3
+4 x −2
3 x −8
3
x 2 3
+2 x −8
a (6 − 3 7) ⎛ ⎝ 3 + 4 7 ⎞ ⎠⎞ b ⎛ ⎝ 3
x − 2 ⎞⎛ 3
⎠⎝ x − 3
⎞ ⎠ .
a ⎛ ⎝ 2−3 11 ⎞ ⎠ (4 − 11) b ⎛ ⎝ 3
x + 1 ⎞⎛ 3
⎠⎝ x + 3
⎞ ⎠ .
⎛ ⎝32− 5 ⎞ ⎠ ⎛ ⎝ 2 +4 5 ⎞ ⎠.
⎛
⎝3 2− 5 ⎞ ⎠ ⎛ ⎝
2 +4 5 ⎞ ⎠
3·2+12 10− 10−4·5
6 + 12 10 − 10−20
−14 + 11 10
⎛ ⎝ ⎛ 5 3 − 7 ⎞ ⎠
⎞⎛
⎝ ⎛ 3 + 2 7 ⎞ ⎠
⎞
⎛ ⎝ ⎛ 6 − 3 8 ⎞ ⎠
⎞⎛
⎝ ⎛ 2 6 + 8 ⎞ ⎠
⎞
Binomial Squares
Product of Conjugates
(a + b) 2 = a 2 +2ab + b 2 (a + b)(a − b) = a 2 − b 2
(a − b) 2 = a 2 −2ab + b 2
a ⎛ ⎝2+ 3 ⎞ ⎠ 2 b ⎛ ⎝4−2 5 ⎞ ⎠ 2 .

2ab
a
b
a ⎛ ⎝ ⎛ 10 + 2 ⎞ ⎠
⎞2
b ⎛ ⎝ ⎛ 1 + 3 6 ⎞ ⎠ ⎞2 .
a ⎛ ⎝ ⎛ 6 − 5 ⎞ ⎠
⎞2
b ⎛ ⎝ ⎛ 9 − 2 10 ⎞ ⎠ ⎞2 .
⎛ ⎝5−2 3 ⎞ ⎠ ⎛ ⎝5+2 3 ⎞ ⎠.

⎛ ⎝ ⎛ 3 − 2 5 ⎞ ⎠
⎞⎛
⎝ ⎛ 3 + 2 5 ⎞ ⎠
⎞
⎛ ⎝ ⎛ 4 + 5 7 ⎞ ⎠ ⎞ (4 − 5 7).

a 82−5 2 b a 72−3 2 b
4 3
5 3 m +2 3 m c 8 4 m −2 m
3 3 3
7 p +2 p c 5 x −3 x
a 35+6 5 b
9 3 a +3 a
3 4
c 5 2z
4
+ 2z
a 45+8 5 b 3 m −4 3 m
c n +3 n
a 32a −4 2a +5 2a b
4
5 3ab
4 4
−3 3ab −2 3ab
a 11b −5 11b +3 11b
4 4 4
b 8 11cd +5 11cd −9 11cd
a 83c +2 3c −9 3c
3 3 3
2 4pq −5 4pq +4 4pq
b
a 35d +8 5d −11 5d b
3
11 2rs
3 3
−9 2rs +3 2rs
a 27 − 75 b
3 3
40 − 320
c 1 4
2 32 + 2 4
3 162
3
a 72 − 98 b 24
c 1 4
2 80 − 2 4
3 405
3
+ 81
a 48 + 27 b
3 3
54 + 128
4
c 6 5− 3 4
2 80
a 45 + 80 b
3 3
81 − 192
c 5 4
2 80 + 7 4
3 405
a 72a 5 − 50a 5 b
4
9 80p 4 4
− 6 405p 4
a 48b 5 − 75b 5 b
3
8 64q 6 3
− 3 125q 6
a 80c 7 − 20c 7 b
4
2 162r 10 4
+4 32r 10
a 96d 9 − 24d 9 b
4
5 243s 6 4
+2 3s 6
3 128y 2 +4y 162 −8 98y 2 3 75y 2 +8y 48 − 300y 2
a (−2 3)(3 18) b a (−4 5)(5 10) b a (5 6)(− 12) b
3 ⎞ 3 ⎞
⎠ ⎠
⎛
⎝ 8 3 4 ⎞⎛ ⎠⎝ −4 18
⎛
⎝ −2 3 9 ⎞⎛ ⎠⎝ 7 9
⎛ 4 ⎞⎛ 4 ⎞
⎝−2 18 ⎠⎝ − 9 ⎠
a (−2 7)(−2 14) b
⎛
⎝ −3 4 8 ⎞ ⎠ ⎛ ⎝ −5 4 6 ⎞ ⎠
a ⎛ ⎝ 412z3 ⎞⎛ ⎠⎝ 39z ⎞ ⎠ b a ⎛ ⎝ 32x3 ⎞⎛ ⎠⎝ 718x 2 ⎞
⎠ b
⎛
⎝ 5 3 ⎞⎛ 3 3x3 ⎠⎝ 3 18x 3 ⎞ ⎛ 3 ⎞⎛ 3
⎠
−6 20a2
⎝ ⎠⎝ −2 16a 3 ⎞
⎠
a ⎛ ⎝ −2 7z3 ⎞⎛ ⎠⎝ 314z 8 ⎞
⎠ b a ⎛ ⎝ 42k5 ⎞⎛ ⎠⎝ −3 32k 6 ⎞
⎠ b
⎛
⎝ 2 4 ⎞⎛ 8y2 4
⎠⎝ −2 12y 3 ⎞ ⎛
⎠ ⎝ − 4 ⎞⎛ 4 6b3 ⎠⎝ 3 8b 3 ⎞
⎠

a 7 ⎛ ⎝5+2 7 ⎞ ⎠ b a 11 ⎛ ⎝8+4 11 ⎞ ⎠ b a 11 ⎛ ⎝−3+4 11 ⎞ ⎠ b
3 ⎞ 3 ⎞ 4
⎠ ⎠
3 ⎛
6 ⎝ 4+ 18
3 ⎛3
3 ⎝ 9 + 18
4 ⎛ 4 ⎞
3 ⎝ 54 + 18 ⎠
a 2 ⎛ ⎝−5 + 9 2 ⎞ ⎠ b
4 ⎛4
2 ⎝ 12
4
+ 24
⎞
⎠
⎛ ⎝7+ 3 ⎞ ⎠ ⎛ ⎝9− 3 ⎞ ⎠ ⎛ ⎝8− 2 ⎞ ⎠ ⎛ ⎝3+ 2 ⎞ ⎠
a ⎛ ⎝9−3 2 ⎞ ⎠ ⎛ ⎝6+4 2 ⎞ ⎠ b a (3−2 7)(5−4 7) b a ⎛ ⎝1+3 10 ⎞ ⎠ ⎛ ⎝5−2 10 ⎞ ⎠ b
3 +1 ⎞ ⎛ 3 −3
⎞ ⎛
⎠ ⎠
⎛3 ⎝ x −3
⎞⎛ ⎠⎝ x
a ⎛ ⎝7−2 5 ⎞ ⎠ ⎛ ⎝4+9 5 ⎞ ⎠ b
3 −5
⎞
⎠
⎛
⎝ x
⎝ x
⎝ 2 3 x +6
⎞⎛3 ⎠⎝ x +1
⎞
⎠
⎛ ⎝
⎛
⎝ 3 3 x +2
⎞⎛3 ⎠⎝ x −2
⎞
⎠
3 + 10 ⎞ ⎠ ⎛ ⎝ 3 +2 10 ⎞ ⎠ ⎛ ⎝ 11 + 5 ⎞ ⎠ ⎛ ⎝ 11 +6 5 ⎞ ⎠
⎛ ⎝27−5 11 ⎞ ⎠ ⎛ ⎝47+9 11 ⎞ ⎠ ⎛ ⎝46+7 13 ⎞ ⎠ ⎛ ⎝86−3 13 ⎞ ⎠ a ⎛ ⎝3+ 5 ⎞ ⎠ 2 b ⎛ ⎝2−5 3 ⎞ ⎠ 2
a ⎛ ⎝4+ 11 ⎞ ⎠ 2 b ⎛ ⎝3−2 5 ⎞ ⎠ 2 a ⎛ ⎝9− 6 ⎞ ⎠ 2 b ⎛ ⎝10 + 3 7 ⎞ ⎠ 2 a ⎛ ⎝5− 10 ⎞ ⎠ 2 b ⎛ ⎝8+3 2 ⎞ ⎠ 2
⎛ ⎝4+ 2 ⎞ ⎠(4− 2) ⎛ ⎝7+ 10 ⎞ ⎠ ⎛ ⎝7− 10 ⎞ ⎠ ⎛ ⎝4+9 3 ⎞ ⎠ ⎛ ⎝4−9 3 ⎞ ⎠
⎛ ⎝1+8 2 ⎞ ⎠ ⎛ ⎝1−8 2 ⎞ ⎠ ⎛ ⎝12 − 5 5 ⎞ ⎠ ⎛ ⎝12 + 5 5 ⎞ ⎠ ⎛ ⎝9−4 3 ⎞ ⎠ ⎛ ⎝9+4 3 ⎞ ⎠
⎛ 3
⎝ 3x
+2 ⎞ ⎠ ⎛ 3
⎝ 3x −2 ⎞ ⎠
⎛ 3
⎝ 4x
+3 ⎞ ⎠ ⎛ 3
⎝ 4x −3 ⎞ ⎠
2
3
27 + 3 4 48 175k4 − 63k 4 5
6
162 + 3 16 128
3 3
24 + /81
1 4
80 − 2 4
2 3 405
4
813
4 4
−4 13−3 13
512c 4 −3 27c 6 80a 5 − 45a 5 3
5
75 − 1 4 48
3 3
21 9 −2 9
3
8 64q 6 3
− 3 125q 6
11 11 −10 11
3 · 21 ⎛ ⎝46 ⎞ ⎠ ⎛ ⎝−18 ⎞ ⎠ ⎛ ⎝ 74 3 ⎞⎛ 3 ⎞
⎠⎝ −3 18 ⎠
⎛ ⎝ 412x5 ⎞
⎠
⎛
⎝ 26x 3 ⎞
⎠
⎛ ⎝ 29 ⎞ ⎠ 2 (−4 17) ⎛ ⎝−3 17 ⎞ ⎠
⎛ ⎝−4 + 17 ⎞ ⎠ ⎛ ⎝−3 + 17 ⎞ ⎠
⎛
⎝ 3 4 ⎞⎛4 8a2 ⎠⎝ 12a 3 ⎞ ⎛ ⎝6−3 2 ⎞2
⎠
⎠

3 ⎛ ⎝4−3 3 ⎞ 3 ⎛ 3
⎠ 3 ⎝ 2 9
3 ⎞
+ 18
⎛ ⎝ 6 + 3 ⎞ ⎠ ⎛ ⎝ 6 +6 3 ⎞ ⎠
⎠
a (− n) 2
n ≥0.
b −( n) 2
n ≥0.
⎛
⎝3+ 2 ⎞ ⎠ 2 .
a
b

30
48 .
x 2 · x 4 .
(7 + 3x)(7 − 3x).
n n
a b b ≠0, n ≥2
n a
b
= n a
b
n and n a n
n = a
b b
a
a
72x3
162x b
3
32x2
3
.
4x 5

72x 3
162x
n
a n
n = a 72x 3
.
b b
162x
18 ·4·x 2 · x
18 ·9· x
4x 2
9
2x
3
b
3
32x 2
3
4x 5
n
a
n = a 3
n 32x 2
.
b b
4x 5
3 8
x 3
2 x
a
50s3
128s b
3
56a
3
.
7a 4
a
75q5
108q b
3
72b2
3
.
9b 5
a 147ab8 b 3
−250mn−2 .
3a 3 b 4 3
2m −2 n 4
a

147ab 8
3a 3 b 4
147ab 8
3a 3 b 4
49b 4
a 2
7b 2
a
b
3
−250mn −2
3
2m −2 n 4
3
−250mn −2
2m −2 n 4
3
−125m 3
n 6
− 5m
n 2
a 162x10 y 2 3
b −128x2 y −1
.
2x 6 y 6 3
2x −1 y 2
a 300m3 n 7
3m 5 n
b
3
−81pq−1 .
3
3p −2 q 5
54x 5 y 3
.
3x 2 y

54x 5 y 3
3x 2 y
54x 5 y 3
3x 2 y
18x 3 y 2
9x 2 y 2 ⋅2x
9x 2 y 2 ⋅
3xy 2x
2x
64x 4 y 5
.
2xyx 3
96a 5 b 4 .
2a 3 b
( a) 2 = a.

a 4 3 b 3 20 c 3 6x .
a
3.
b
5.
c
6x.
a 5 3 b 3 32 c 2 2x .

a 6 5 b 7 18 c 5 5x .
a 1 3
b 7 3 24
6
c
3 .
3
4x
a
3
6 2 ,
b

3
3 2 .
3
3 3
=3.
c
3
2·x 2 .
a 1 3
b 5 3
7
12
c
5 .
3
9y
a 1 3
b 3 3 20
2
c
2 .
3
25n
a 1 4
b 5 4
2
64
c
2 .
4
8x
a

4
2 3 ,
b
4
2 2 .
4
2 4
=2.
c
4
2·x 3 .

a 1 4
b 3 4
3
64
c 3 .
4
125x
a 1 4
b 7
4 128
5
c
4 4x
4
(a − b)(a + b)
⎛
⎝2− 5 ⎞ ⎠ ⎛ ⎝2+ 5 ⎞ ⎠
a 2 − b 2 2 2 − ⎛ ⎝ 5 ⎞ ⎠ 2
4−5
−1
5
2− 3 .
3
1 − 5 .
2
4 − 6 .

3
u − 6 .
5
x + 2 .
10
y − 3 .
x + 7
x − 7 .
p + 2
p − 2 .

q − 10
q + 10

a 128
72 b 3
128
3
54
a 48
75 b 3
81
3
24
a
3
200m5
98m b 54y2
3
2y 5
a
108n7 b
243n 3
3
54y
3
16y 4
a
75r3 b
108r 7
3
24x7
3
81x 4
a
196q
484q 5 b
3
16m4
3
54m
a 108p5 q 2 3
b −16a4 b −2
3
3p 3 q 6 3
2a −2 a 98rs10 b −375y4 z −2
b
2r 3 s 4 3
3y −2 z 4
a
320mn−5 b
45m −7 n 3
3
16x4 y −2
3
−54x −2 y 4
a 810c−3 d 7
1000cd −1 b
3
24a7 b −1
3
−81a −2 b 2
56x 5 y 4
2xy 3 3ab 3
72a 3 b 6
3
48a 3 b 6
3
3a −1 b 3
3
162x −3 y 6
3
2x 3 y −2
a 10 6 b
4 27 c 10 5x
a 8 3 b 7 40 c 8 2y
a 6 7 b 8 45 c 12
3p
a 4 5 b 27
80 c 18 6q
a 1 3
b 5 3 24
5
c
4 3
36a
a 1 3
b 5 3 32
3
c
7 3
49b
a 1 3
b 7 3
11
54
c 3 3
3x 2
a 1 3
b 3
3 128
13
c 3 3
6y 2
a 1 4
b 5 4 7
32
c 4 4
4x 2
a 1 4
b 9 4
4
32
c 6 4
9x 3
a 1 4
b 25
4
9
128
c
6 4
27a
a 1 4
b 27
4
8
128
c 16
4
64b 2

8
1− 5
7
2− 6
6
3− 7
5
4− 11
3
m − 5
5
n − 7
2
x − 6
7
y + 3
r + 5
r − 5
s − 6
s + 6
x + 8
x − 8
m − 3
m + 3
a
27
3
b
27
5
c
2x −3
7 x
3
3
x
3
x
a
b

⎛
⎝y
− 3 ⎞2 ⎠ .
2x − 5 = 0.
n 2 − 6n + 8 = 0.
For a ≥0, ( n a) n = a.
5n −4−9=0.

3m + 2 − 5 = 0.
10z + 1 − 2 = 0.
9k −2+1=0.
2r − 3 + 5 = 0.

7s − 3 + 2 = 0.
(a + b) 2 = a 2 +2ab + b 2
(a − b) 2 = a 2 −2ab + b 2
p −1+1= p.
p =1, p =2.
x − 2 + 2 = x.
y − 5 + 5 = y.
⎛ 3 ⎝ a ⎞ 3
⎠ = a

3
5x +1+8=4.
3
5x +1+8=4
3
5x +1
=−4
3
⎛3 ⎞
⎝ 5x +1 ⎠ = (−4) 3
5x + 1 = −64
5x = −65
x = −13
x = −13.
3
4x − 3 + 8 = 5
3
6x − 10 + 1 = −3
(a m ) n = a m · n ,
⎛ 1
⎜x
⎞ 2
2
⎝ ⎠ ⎟ = x,
⎛ 1 ⎞ 3
3 ⎜x
⎝ ⎠ ⎟ = x
1
1
x
2 = x x
3 3
= x .
1
(3x −2)
4 +3=5.
1
(3x −2)
4 +3=5

1
(3x −2)
4 =2
⎛ 1
⎜(3x ⎞ 4
4
−2)
⎝ ⎠ ⎟ = (2) 4
3x −2=16
3x =18
x =6
x =6.
1
(9x + 9)
4 − 2 = 1.
1
(4x − 8)
4 + 5 = 7.
r +4− r +2=0.
r +4− r +2=0
r +4= r −2
⎛
⎝ r +4 ⎞ ⎠ 2 = (r −2) 2
r +4=r 2 −4r +4
0=r 2 −5r
0=r(r −5)
0=r 0=r −5

=0 r =5
r =0
m + 9 − m + 3 = 0.
n + 1 − n + 1 = 0.
3 3x −5−8=4.
3 3x −5−8=4
33x −5=12
3x −5=4
⎛
⎝ 3x −5 ⎞2 ⎠ = (4) 2
3x −5=16
3x =21
x =7
x =7.
2 4a + 4 − 16 = 16.

3 2b + 3 − 25 = 50.
3 3
4x −3= 3x +2.
3
4x −3
3
= 3x +2
3
⎛3 ⎞
⎝ 4x −3 ⎠ = ⎛ 3
3 ⎞
⎝ 3x +2 ⎠
4x −3=3x +2
x −3=2
x =5
The solution is x =5.
3
5x − 4
3
= 2x + 5.
3
7x + 1
3
= 2x − 5.
m +1= m +9.

3 − x = x − 3.
x + 2 = x + 16.
(a + b) 2 = a 2 +2ab + b 2
(a − b) 2 = a 2 −2ab + b 2 .
q −2+3= 4q +1.

q −2.
q =6 q =2.
x − 1 + 2 = 2x + 6
x + 2 = 3x + 4

t = h 4 .
h =64
t =
h
4
t =

t = h 4
t =
h
4
s =
24d
s =
24d
s =
s = 24d

s =
24d

5x −6=8 4x −3=7 5x +1=−3
3y −4=−2 3
2x
3
=−2 4x −1
2m −3−5=0 2n −1−3=0 6v −2−10=0
12u +1−11=0 4m +2+2=6 6n +1+4=8
=3
2u −3+2=0 5v −2+5=0 u −3+3=u
v −10+10=v r −1= r −1 s −8= s −8
3
6x +4
3
=4 11x +4
3
=5 4x +5−2=−5
3
9x −1−1=−5
1
1
(6x +1)
−3=4 (3x −2)
+1=6
1
1
1
(8x +5)
+2=−1 (12x −5)
+8=3 (12x −3)
−5=−2
1
(5x −4)
4 +7=9
x +1− x +1=0 y +4− y +2=0
z + 100 − z = −10 w +25− w =−5 32x −3−20=7
25x +1−8=0 28r +1−8=2 37y +1−10=8
3u +7= 5u +1 4v +1= 3v +3 8+2r = 3r +10
10 + 2c = 4c +16 3
5x −1
3
= x +3
3
8x −5
3
= 3x +5
3
2x 2 +9x −18
3
= x 2 +3x −2
3
x 2 − x +18
3
= 2x 2 −3x −6
a +2= a +4
r +6= r +8 u +1= u +4 x +1= x +2
a +5− a =1 −2 = d −20− d 2x +1=1+ x
3x +1=1+ 2x −1 2x −1− x −1=1 x +1− x −2=1

x +7− x −5=2 x +5− x −3=2
s = A
s = A
t =
4 h
t =
4 h
s = 24d
s = 24d
x +1=0
a r +4− r +2=0. b
a
b

1 − 2x ≥ 0.
f (x) = 3x − 4, f (2), f (−1), f (0).
f (x) = x.
f (x) = x.
f (x) =
3
x .
f (x) =
2x −1, a f (5) b f (−2).
a
f (x) =
2x −1
f (5), x. f (5) = 2·5−1
f (5) = 9
f (5) =3

f (x) =
2x −1
f (−2), x. f (−2) = 2(−2) −1
f (−2) = −5
x =−2.
f (x) = 3x − 2, a f (6) b f (0).
g(x) = 5x + 5, a g(4) b g(−3).
g(x) =
3
x −6, a g(14) b g(−2).
a
g(x) =
3
x −6
g(14), x.
g(14) =
g(14) = 3 8
3
14−6
g(14) =2
b
g(x) =
3
x −6
g(−2), x.
g(−2) =
g(−2) =
3
−2−6
3
−8
g(−2) =−2
3
g(x) = 3x − 4, a g(4) b g(1).
3
h(x) = 5x − 2, a h(2) b h(−5).

f (x) =
4
5x −4, a f (4) b f (−12)
a
f (x) =
4
5x −4
f (4), x.
4
f (4) = 5·4−4
4
f (4) = 16
f (4) =2
b
f (x) =
4
5x −4
f (−12), x.
f (−12) =
f (−12) =
4
5(−12) −4
4
−64
x = −12.
4
f (x) = 3x + 4, a f (4) b f (−1).
4
g(x) = 5x + 1, a g(16) b g(3).
n
a
a ≥0, n a
a

f (x) =
3x −4.
f (x) = 3x −4
3x −4≥0
3x ≥4
x ≥ 4 3
f (x) = 3x −4 x ≥ 4 3 ⎡ ⎣ 4 3 ,∞⎞ ⎠ .
f (x) = 6x − 5.
f (x) = 4 − 5x.
g(x) = 6
x −1 .
g(x) = 6
x −1
6
x −1
x −1>0
x −1 > 0
x > 1
x −1=0
x −1 = 0
x = 1sox ≠ 1 in the domain.
x >1 (1, ∞).
f (x) = 4
x + 3 .

h(x) = 9
x − 5 .
3
f (x) = 2x 2 +3.
3
f (x) = 2x 2 +3
(−∞, ∞).
3
f (x) = 2x 2 +3
(−∞, ∞).
3
f (x) = 3x 2 − 1.
3
g(x) = 5x − 4.
f (x) = x,
x ≥0 [0, ∞).
f (x) = x.
[0, ∞).
f (x) = x +3,
a b c
a x +3≥0, x ≥−3.
x ≥−3 [−3, ∞).
b [−3, ∞)

c [0, ∞).
f (x) = x + 2, a b c
f (x) = x − 2, a b c
f (x) = x 3 3
f (x) = x .
f (x) =
3
x , a b c
a
(−∞, ∞)
b (−∞, ∞)
c (−∞, ∞).
3
f (x) = − x ,
a b c

3
f (x) = x − 2,
a b c

f (x) = 4x −4, a f (x) = 6x −5, a
f (5) b f (0).
f (5) b f (−1).
g(x) = 6x +1, a
g(4) b g(8).
g(x) = 3x +1, a
g(8) b g(5).
G(x) = 5x −1, a
G(5) b G(2).
F(x) = 3−2x, a
F(1) b F(−11).
G(x) = 4x +1, a
G(11) b G(2).
F(x) = 8−4x, a
F(1) b F(−2).
3
g(x) = 2x −4, a
g(6) b g(−2).
3
g(x) = 7x −1, a
g(4) b g(−1).
3
h(x) = x 2 −4, a
h(−2) b h(6).
3
h(x) = x 2 +4, a
h(−2) b h(6).
4
f (x) = 2x 3
f (2).
, a f (0) b
4
f (x) = 3x 3
f (3).
, a f (0) b
4
g(x) = 4−4x
g(−3).
, a g(1) b
4
g(x) = 8−4x, a g(−6)
b g(2).
f (x) = 3x −1 f (x) = 4x −2 g(x) = 2−3x
g(x) = 8−x h(x) = 5
x −2
h(x) = 6
x +3
f (x) = x +3
x −2
f (x) =
x −1
x +4
g(x) =
3
8x −1
3
g(x) = 6x +5
f (x) =
3
4x 2 −16
f (x) =
3
6x 2 −25
4
F(x) = 8x +3
F(x) =
4
10−7x
G(x) =
5
2x −1
G(x) =
5
6x −3

a b c
f (x) = x +1 f (x) = x −1 g(x) = x +4
g(x) = x −4 f (x) = x +2 f (x) = x −2
g(x) =2 x g(x) =3 x f (x) = 3−x
f (x) = 4−x g(x) =− x g(x) =− x +1
3
f (x) = x +1
f (x) =
3
x −1
3
g(x) = x +2
g(x) =
3
x −2
f (x) =
3
x +3 f (x) =
3 x −3
g(x) =
3
x
3
g(x) =− x
3
f (x) =2 x
3
f (x) =−2 x
y =
3
x
a
b

i
−4, − 7, 0. 5 – , 7 , 3, 81, a b c
3
(x − 3)(2x + 5).
5
5 − 3 .
−1,
5.
i
i 2 =−1ori =
−b =
b i
−1

−b = ib −b = b i
a −25 b −7 c −12.
a
−25
25 i
5i
b
−7
7i
i
c
−12
12 i
12. 23i
a −81 b −5 c −18.
a −36 b −3 c −27.

a + bi,
b =0, a + bi a +0·i = a,
b ≠0, a + bi
a =0, a + bi 0+bi = bi,
a + bi
b =0
a +0·i
a
b ≠0 a + bi
a =0
0+bi
bi
a + bi, −b = bi b >0.
−12 + −27.
−12 + −27
12 i + 27i
23i +3 3i
53i
−8 + −32.

−27 + −48.
a (4−3i) + (5+6i) b (2−5i) − (5−2i).
a
(4−3i) + (5+6i)
(4+5) + (−3i +6i)
9+3i
b
(2−5i) − (5−2i)
2−5i −5+2i
2−5−5i +2i
−3 − 3i
a (2 + 7i) + (4 − 2i) b (8 − 4i) − (2 − i).
a (3 − 2i) + (−5 − 4i) b (4 + 3i) − (2 − 6i).
2i(7−5i).
2i(7−5i)
14i −10i 2
i 2 .
14i −10(−1)
14i +10
10 + 14i

4i(5 − 3i).
−3i(2 + 4i).
(3+2i)(4−3i).
(3+2i)(4−3i)
12 − 9i +8i −6i 2
i 2
12 − i −6(−1)
12 − i +6
18 − i
(5 − 3i)(−1 − 2i).
(−4 − 3i)(2 + i).
(3 + 2i) 2
(a + b) 2 = a 2 +2ab + b 2 .
i 2 .
(−2 − 5i) 2 .
(−5 + 4i) 2 .
−b = bi.

−36 · −4.
−36 · −4
−b = bi. 36 i · 4i
6i ·2i
12i 2
i 2
−12
−49 · −4.
−36 · −81.
⎛ ⎝3 − −12 ⎞ ⎠ ⎛ ⎝5 + −27 ⎞ ⎠.
⎛
⎝3 − −12 ⎞ ⎠ ⎛ ⎝5 + −27 ⎞ ⎠
−b = bi.
⎛
⎝3−2 3i ⎞ ⎠ ⎛ ⎝5+3 3i ⎞ ⎠
15 + 9 3 i −10 3i −6·3i 2
i 2 .
15 − 3 i −6·(−3)
33 − 3 i
(4 − −12) ⎛ ⎝ ⎛ 3 − −48 ⎞ ⎠ ⎞ .
⎛ ⎝ ⎛ −2 + −8 ⎞ ⎠
⎞⎛
⎝ ⎛ 3 − −18 ⎞ ⎠ ⎞ .
(a − b), (a + b).

a + bi, a − bi.
a + bi, a − bi.
(3−2i)(3+2i).
(3−2i)(3+2i)
9+6i −6i −4i 2
i 2 .
9−4(−1)
(4 − 3i) · (4 + 3i).
(−2 + 5i) · (−2 − 5i).
(a − b)(a + b) = a 2 − b 2 .
i 2 −1.
(a − bi)(a + bi)
a 2 − (bi) 2
a 2 − b 2 i 2
a 2 − b 2 (−1)
a 2 + b 2
(a − bi)(a + bi) = a 2 + b 2
(a − bi)(a + bi) = a 2 + b 2

(8 − 2i)(8 + 2i).
(a − bi)(a + bi) = a 2 + b 2 .
(3 − 10i)(3 + 10i).
(−5 +4i)(−5 − 4i).
4+3i
3−4i .
2 + 5i
5 − 2i .
1 + 6i
6 − i .

−3
5+2i .
−3
5+2i
−3(5 − 2i)
(5 + 2i)(5 − 2i)
−15 + 6i
5 2 +2 2
−15 + 6i
29
− 15
29 + 6 29 i
4
1 − 4i .
−2
−1 + 2i .
5+3i .
4i

5+3i
4i
5+3i
0+4i
(5+3i)(0−4i)
(0+4i)(0−4i)
(5+3i)(−4i)
(4i)(−4i)
−20i −12i 2
−16i 2
i 2 . −20i +12
16
12
16 − 20
16 i
3
4
− 5 4 i
3 + 3i.
2i
2 + 4i.
5i
i i
i 1 i 2 i 3 i 4
i −1 i 2 · i i 2 · i 2
−1·i (−1)(−1)
−i 1
i 5 i 6 i 7 i 8
i 4 · i i 4 · i 2 i 4 · i 3 i 4 · i 4
1·i 1·i 2 1·i 3 1·1
i i 2 i 3 1
−1 −i
i 1 = i i 5 = i
i 2 = −1 i 6 = −1
i 3 = −i i 7 = −i
i 4 = 1 i 8 = 1

i n = ⎛ ⎝i 4⎞ ⎠q
· i r ,
57 = 4 · 14 + 1. i 57 = ⎛ ⎝1 4⎞ ⎠14
· i
1
i 86 .
i 86
i 86
⎛
⎝ 1 4⎞ 21
⎠ · i
2
i n = ⎛ ⎝i 4⎞ q
⎠ · i
r
(1) 21 · (−1)
–1
i 75 .
i 92 .

a −16 b −11
c −8
a −121 b −1 c −20
a −100 b −13 c −45
a −49 b −15 c −75
−75 + −48 −12 + −75 −50 + −18
−72 + −8 (1+3i) + (7+4i) (6+2i) + (3−4i)
(8−i) + (6+3i) (7−4i) + (−2−6i) (1−4i) − (3−6i)
(8−4i) − (3+7i) (6+i) − (−2−4i) (−2 + 5i) − (−5 + 6i)
⎛ ⎝5 − −36 ⎞ ⎠ + ⎛ ⎝2 − −49 ⎞ ⎠
⎛ ⎝−3 + −64 ⎞ ⎠ + ⎛ ⎝5 − −16 ⎞ ⎠
⎛
⎝−7 − −50 ⎞ ⎠ − ⎛ ⎝−32 − −18 ⎞ ⎠
⎛
⎝−5 + −27 ⎞ ⎠ − ⎛ ⎝−4 − −48 ⎞ ⎠
4i(5−3i) 2i(−3 + 4i) −6i(−3−2i)
−i(6+5i) (4+3i)(−5 + 6i) (−2−5i)(−4 + 3i)
(−3+3i)(−2−7i) (−6−2i)(−3 − 5i)
(3+4i) 2 (−1+5i) 2 (−2 − 3i) 2
(−6 − 5i) 2
−25 · −36 −4 · −16 −9 · −100
−64 · −9 (−2 − −27) ⎛ ⎝4 − −48 ⎞ ⎠ ⎛ ⎝5 − −12 ⎞ ⎠ ⎛ ⎝−3 + −75 ⎞ ⎠
⎛ ⎝2+ −8 ⎞ ⎠ ⎛ ⎝−4 + −18 ⎞ ⎠
⎛ ⎝5 + −18 ⎞ ⎠ ⎛ ⎝−2 − −50 ⎞ ⎠
(2−i)(2+i)

(4−5i)(4+5i) (7−2i)(7+2i) (−3−8i)(−3 + 8i)
(7−i)(7+i) (6−5i)(6+5i) (9−2i)(9+2i)
(−3−4i)(−3 + 4i)
3+4i
4−3i
5−2i
2+5i
2+i
3−4i
3−2i
6+i
3
2−3i
2
4−5i
−4
3−2i
−1
3+2i
1+4i
3i
4+3i
7i
−2−3i
4i
−3 − 5i
2i
i 41 i 39 i 66
i 48 i 128 i 162
i 137 i 255
−7 · −7
49
7
3
−64 =8i −64
=−4.

a
b 1 − 10,

i i = −1.
a + bi,
m
n = m, n ≥0.
m,
b n = a,
n a.
n
a
a ≥0, n a
a

n ≥2
n
a m n
n a n b, n ≥2
n
ab = n n
a · b
n n n
a · b = ab
n n
a b b ≠0, n ≥2
n a = n b
a n n a n
n = a
b b b
1
n
a
n a
m
a n
1
n ≥2, a
n =
n a.
m
a n
= ( n a) m m
a n n
= a m
a m · a n = a m + n
(a m ) n = a m · n
(ab) m = a m b m
am
a n = am − n , a ≠0
a 0 =1, a ≠0
⎛ ⎝ a m
⎞
b⎠
= am
b
m, b ≠0
a −n = 1 a
n, a ≠0
n a n b, n ≥2

n
ab
= n n
a · b
n n n
a · b = ab
Binomial Squares Product of Conjugates
(a + b) 2 = a 2 +2ab + b 2 (a + b)(a − b) = a 2 − b 2
(a − b) 2 = a 2 −2ab + b 2
n n
a b b ≠0, n ≥2
n a = n b
a n n a n
n = a
b b b
(a + b) 2 = a 2 +2ab + b 2
(a − b) 2 = a 2 −2ab + b 2
t =
4 h.
s = 24d.

n
a
a ≥0,
n a
a

3
a 225 b −16 a − 169 b −8 a 8
4
b 81
5
c 243
3
a −512
4
b −81
5
c −1
3
a 68 b 84
3
a 37 b 84
4
c 125
3
a a 3
a a 14
7
b b 7
b w 24
4
a m 8
5
b n 20
a 121m 20
b −64a 2
3
a 216a 6
5
b 32b 20
a 144x 2 y 2
b 169w 8 y 10
3
c 8a 51 b 6
125 675 3
a 625
6
b 128
a a 23
3
b b 8
8
c c 13
a 80s 15
5
b 96a 7
6
c 128b 7
a 96r 3 s 3
3
b 80x 7 y 6
4
c 80x 8 y 9
5
a −32
8
b −1
a 8+ 96
b 2+ 40
2

a 72
98 b 3
24 4
c
81 96
6 c
a y4
y 8 b
5
u21
u 11
6
v30
v 12
300m 5
64
a
28p7
q 2
3
b 81s8
t 3
c
4
64p15
q 12
a
27p2 q
108p 4 q 3
3
b 16c5 d 7
250c 2 d 2
6
c 2m9 n 7
128m 3 n
a
80q5
5q
3
b −625
3
5
4
c 80m7
4
5m
1
n
a
1 1 1
a r 2
b s 3 4
c t
4
a 21p b 8q
6
c 436r
1
4
a 625
1
5
b 243
1
5
c 32
1
3
a (−1,000)
1
3
b −1,000
c (1,000) − 1 3
1
5
a (−32)
b (243) − 1 5
1
3
c −125
m
a n
4
a r 7
b ⎛ 3
5
⎝ 2pq ⎞
⎠
4
c ⎛ ⎝ 12m 3
⎞
7n ⎠

3
2
a 25
b 9 − 3 2
2
3
c (−64)
3
2
a −64
b −64 − 3 2
3
2
c (−64)
5
a 6 2 1
3
2
·6
a a 4 · a − 1 4
3
b ⎛ ⎝b 15⎞ 5
a − 10 4
⎠
1
2 ⎛ 2
c w 7
b
⎜27b
3 c − 5 3
⎞
2 ⎟
9
⎜
7
w
⎝ b − 73 1 ⎟
2
c ⎠
a 72−3 2
a 11b −5 11b +3 11b
3 3
4 4 4
b 7 p +2 p
b 811cd +5 11cd −9 11cd
3
c 5 x −3
3 x
a 48 + 27
3 3
b 54 + 128
4
c 65− 3 4
2 80
a 80c 7 − 20c 7
4
b 2 162r 10 4
+4 32r 10
375y 2 +8y 48 − 300y 2
a ⎛ ⎝56 ⎞ ⎠(− 12)
a ⎛ ⎝ 32x3 ⎞⎛ ⎠⎝ 718x 2 ⎞
⎠
b ⎛ 4 ⎞⎛ 4 ⎞
⎝−2 18 ⎠⎝ −9 ⎠
b ⎛ 3 ⎞⎛ 3
−6 20a2
⎝ ⎠⎝ −2 16a 3 ⎞
⎠
a 11 ⎛ ⎝8+4 11 ⎞ ⎠
a (3−2 7)(5−4 7)
3 ⎛3
3 ⎞
b 3 ⎝ 9 + 18
b ⎛ 3
⎠
⎝ x −5
⎞⎛3 ⎠⎝ x −3
⎞
⎠
⎛
⎝2 7−5 11 ⎞ ⎠ ⎛ ⎝47+9 11 ⎞ ⎠

a ⎛ ⎝4+ 11 ⎞ ⎠ 2
⎛
⎝7+ 10 ⎞ ⎠ ⎛ ⎝7− 10 ⎞ ⎠
⎛3
⎝ 3x
+2 ⎞ ⎠ ⎛ 3
⎝ 3x −2 ⎞ ⎠
b ⎛ ⎝3−2 5 ⎞ ⎠ 2
a 48
75
3
b 81
3
24
a 320mn−5
45m −7 n 3
3
b
16x4 y −2
3
−54x −2 y 4
a 8 3 b 7 40 c 8 2y
a 1 3
b 7 3
11
54
c 3 3
3x 2
a 1 4
b 9 4
4
32
c 6 4
9x 3
7
2− 6
5
n − 7
x + 8
x − 8
4x −3=7 5x +1=−3 3
4x −1=3
u −3+3=u 3
4x +5−2=−5
1
(8x +5)
3 +2=−1
y +4− y +2=0 28r +1−8=2
10 + 2c = 4c +16
3
2x 2 +9x −18
3
= x 2 +3x −2
r +6= r +8
x +1− x −2=1

s = A
s = 24d
g(x) = 6x +1, G(x) = 5x −1,
a g(4)
b g(8)
a G(5)
b G(2)
3
h(x) = x 2 −4,
a h(−2)
b h(6)
4
g(x) = 4−4x,
a g(1)
b g(−3)
g(x) = 2−3x F(x) = x +3
x −2
4
F(x) = 10−7x
3
f (x) = 4x 2 −16
a b c
g(x) = x +4 g(x) =2 x
3
f (x) = x −1
3
f (x) = x +3
a −100
b −13
c −45

−50 + −18 (8−i) + (6+3i) (6+i) − (−2−4i)
⎛
⎝−7 − −50 ⎞ ⎠ − ⎛ ⎝−32 − −18 ⎞ ⎠
(−2−5i)(−4 + 3i) −6i(−3−2i) −4 · −16
⎛
⎝5 − −12 ⎞ ⎠ ⎛ ⎝−3 + −75 ⎞ ⎠
(−2−3i) 2
(9−2i)(9+2i)
2+i
3−4i
−4
3−2i
i 48 i 255

3
125x 9
169x 8 y 6 3
72x 8 y 4
45x 3 y 4
180x 5 y 2
a 256 − 1 3
4 2
b −49
−45
x − 14 5
4
· x
x − 3 4
⎛ 2
⎜8x
⎜
⎝ x − 73 y
3 y − 5 2
1
2
1
3 48x 5 − 75x 5 27x 2 −4x 12 + 108x 2
⎞
⎟
⎟
⎠
212x 5 ·3 6x 3 3 ⎛3
4 ⎝ 16
− 3 ⎞
6
⎠
⎛
⎝4−3 3 ⎞ ⎠ ⎛ ⎝5+2 3 ⎞ ⎠
3
128
3
54
245xy −4
1
3
45x −4 y 3 5
3
2+ 3
4+i
3−2i
−4 · −9 −4i(−2−3i)
i 172
2x +5+8=6 x +5+1=x
3
2x 2 −6x −23
3
= x 2 −3x +5
a b c
g(x) = x +2

ax 2 = k
a(x – h) 2 = k
128.

32
5
9x 2 − 12x + 4
a ≠0
ax 2 = k
x 2 =9
x 2 −9=0
(x −3)(x +3) =0
x −3=0 x −3=0
x =3 x =−3
x =4,x =−4 x =5,x =−5.
If n 2 = m, then n is a square root of m.
x = k or x =− k or x =± k.
k
x =± k.

x 2 =9
x =± 9
x =±3
So x =3 or x =−3.
x 2 =7
x = 7, x =− 7
7
x 2 −50=0.
x 2 − 48 = 0.
y 2 − 27 = 0.

3z 2 = 108.
3z 2 = 108
3z 2
3 = 108
3
z 2 =36
z =± 36
z =±6
z =6, z =−6
2x 2 = 98.
5m 2 = 80.
x 2 = k k

x 2 = −72
x = ± −72
x =± 72i
x =±62i
x =6 2i, x =−6 2i
c 2 + 12 = 0.
q 2 + 24 = 0.
2 3 u2 + 5 = 17.
2
3 u2 +5=17
3 2

1 2 x2 + 4 = 24.
3 4 y2 − 3 = 18.
2x 2 − 8 = 41.
2
5r 2 − 2 = 34.
3t 2 + 6 = 70.

4 ⎛ ⎝y −7 ⎞ ⎠2 = 48.
4 ⎛ ⎝y −7 ⎞ ⎠2 =48
⎛
⎝y −7 ⎞ ⎠2 =12
y −7= ± 12
y −7= ±2 3
y. y =7±2 3
y =7+2 3, y =7−2 3
3(a − 3) 2 = 54.
2(b + 2) 2 = 80.
⎛ ⎝ x − 1 ⎞
3⎠2 = 5.
9

⎛
⎝ x − 1 ⎞
3⎠2 = 5
9
x − 1 3 =± 5 9
x − 1 3 =± 5 9
x − 1 3 =± 5 3
x x = 1 3 ± 5 3
x = 1 3 + 5 3 , x = 1 3 − 5 3
⎛
⎝ x − 1⎞ 2
2⎠ = 5 4 .
⎛
⎝ y + 3⎞ 2
4⎠ =
16 7 .
2(x −2) 2 + 3 = 57.
2(x −2) 2 +3=57
2(x −2) 2 =54
(x −2) 2 =27
x −2= ± 27
x −2= ±3 3
x x =2±3 3
x =2+3 3, x =2−3 3

5(a − 5) 2 + 4 = 104.
3(b + 3) 2 − 8 = 88.
(2x −3) 2 = −12.
(2x −3) 2 = −12
2x − 3 = ± −12
2x −3= ±2 3i
2x =3±2 3i
x =
3±2 3i
2
x = 3 2 ± 23i
2
x = 3 2 ± 3i
x = 3 2 + 3i, x = 3 2 − 3i
(3r + 4) 2 = −8.
(2t − 8) 2 = −10.
4n 2 +4n + 1 = 16.
4n 2 +4n +1=16
(2n +1) 2 =16

2n +1= ± 16
2n +1= ±4
n
2n =−1±4
2n
2
= −1±4
2
n = −1 ± 4
2
n = −1 + 4
2
n = −1 − 4
2
n = 3 2 n =−5 2
9m 2 − 12m +4= 25.
16n 2 +40n + 25 = 4.

a 2 =49 b 2 = 144 r 2 −24=0
t 2 −75=0 u 2 − 300 = 0 v 2 −80=0
4m 2 =36 3n 2 =48 4
3
x 2 =48
5
3
y 2 =60 x 2 +25=0 y 2 +64=0
x 2 +63=0 y 2 +45=0 4
3
x 2 + 2 = 110
2
3
y 2 −8=−2 2
5
a 2 +3=11 3
2
b 2 −7=41
7p 2 +10=26 2q 2 +5=30 5y 2 −7=25
3x 2 −8=46
(u −6) 2 =64 (v +10) 2 = 121 (m −6) 2 =20
(n +5) 2 =32
⎛ ⎝ r − 1 ⎞
2⎠2 = 3 ⎛ 4 ⎝ x + 1 2
⎞
5⎠
=
25
7
⎛ ⎝ y + 2 ⎞
3⎠2 = 8 ⎛ 81 ⎝ t − 5 2
⎞
6⎠
= 11
25
(a −7) 2 +5=55
(b −1) 2 −9=39 4(x +3) 2 −5=27 5(x +3) 2 −7=68
(5c +1) 2 = −27 (8d −6) 2 = −24 (4x −3) 2 + 11 = −17
⎛ ⎝2y +1 ⎞ ⎠2 − 5 = −23 m 2 −4m +4=8 n 2 +8n +16=27
x 2 −6x +9=12 y 2 +12y +36=32 25x 2 −30x +9=36
9y 2 +12y +4=9 36x 2 −24x +4=81 64x 2 + 144x +81=25

2r 2 =32 4t 2 =16
(a −4) 2 =28 (b +7) 2 =8
9w 2 −24w +16=1 4z 2 +4z +1=49
a 2 −18=0 b 2 − 108 = 0
⎛ ⎝ p − 1 ⎞
3⎠2 = 7 ⎛ 9 ⎝ q − 3 2
⎞
5⎠
= 3 4
m 2 +12=0 n 2 +48=0.
u 2 −14u +49=72 v 2 +18v +81=50
(m −4) 2 +3=15 (n −7) 2 −8=64
(x +5) 2 =4 ⎛ ⎝y −4 ⎞ ⎠2 =64
6c 2 +4=29 2d 2 −4=77
(x −6) 2 +7=3 ⎛ ⎝y −4 ⎞ ⎠2 +10=9
(x +2) 2 =16
a
b

x 2 + bx + c =0
ax 2 + bx + c =0
(x + 9) 2 .
y 2 − 14y + 49.
5n 2 + 40n + 80.
⎛
⎝y −7 ⎞ ⎠2 = 12
y −7 = ± 12
y −7 = ±2 3
y = 7±2 3
(x − k) 2
x 2 −10x +25 = 18
(x −5) 2 = 18
(x +9) 2 ⎛ ⎝y −7 ⎞ 2
⎠
⎛
(x +9)(x +9) ⎝y −7 ⎞ ⎠ ⎛ ⎝y −7 ⎞ ⎠
x 2 +9x +9x +81 y 2 −7y −7y +49
x 2 +18x +81 y 2 −14y +49

1 (6).
2
x 2 + bx.
⎛1 ⎞ 2
⎝ b 2 ⎠ ,
⎛1 ⎞ 2
⎝ b 2 ⎠
a x 2 −26x b y 2 −9y c n 2 + 1 2 n

a
x
Find ⎛ ⎝ 1 2 b⎞ ⎠2
.
⎛
⎝ 1 2 · 2
(−26)⎞ ⎠
(13) 2
169
b
y −9
Find ⎛ ⎝ 1 2 b⎞ ⎠2
.
⎛
⎝ 1 2 · 2
(−9)⎞ ⎠
⎛
⎝ −9 2
⎞
2⎠
81
4
81 4
c
n 1 2 .

Find ⎛ ⎝ 1 2 b⎞ ⎠2
.
⎛
⎝ 1 2 · 12
2
⎞
⎠
⎛
⎝ 1 2
⎞
4⎠
1
16
1 16
a a 2 − 20a b m 2 − 5m c p 2 + 1 4 p
a b 2 − 4b b n 2 + 13n c q 2 − 2 3 q
x 2 + bx + c =0
x 2 +8x = 48.

x 2 + 4x = 5.
y 2 − 10y = −9.
x 2 + bx + c =0
⎛1 ⎝ · b ⎞ 2 ⎠ ⎞2 ,

x 2 +4x = −21.
⎛
⎝ 1 2 (4)⎞ ⎠2
=4
y 2 − 10y =−35.
z 2 + 8z = −19.
y 2 −18y =−6.
−18
⎛
⎝ 1 2 (−18)⎞ ⎠2
=81

y
y 2 −18y
−6.
x 2 − 16x = −16.
y 2 + 8y = 11.
x 2 +10x + 4 = 15.
⎛
⎝ 1 2 (10)⎞ ⎠2
=25

a 2 + 4a + 9 = 30.
b 2 + 8b − 4 = 16.
n 2 =3n + 11.
3n
−3
⎛
⎝ 1 2 (−3)⎞ ⎠2
= 9
4
9 4
p 2 = 5p + 9.

q 2 = 7q − 3.
a · b =0,
(x −3)(x +5) =9.
⎛
⎝ 1 2 ·(2)⎞ ⎠2
=1
(c − 2)(c + 8) = 11.
(d − 7)(d + 3) = 56.
3x 2 −12x −15=0.
x 2 x 2

⎛
⎝ 1 2 (−4)⎞ ⎠2
=4
2m 2 + 16m + 14 = 0.
4n 2 − 24n − 56 = 8.
2x 2 −3x = 20.
x 2

x 2
− 3 2
⎛
⎝ 1 ⎛
2⎝ − 3 ⎞
2⎠ ⎞ ⎠2
= 9
16
9 16
3r 2 − 2r = 21.
4t 2 + 2t = 20.

ax 2 + bx + c =0
a
⎛1 ⎝ · b ⎞ 2 ⎠ ⎞2 ,
3x 2 +2x =4.
x 2
2 3
⎛
⎝ 1 2 · 23
⎞
⎠2
= 1
9
1 9

4x 2 + 3x = 2.
3y 2 − 10y = −5.

a m 2 −24m b x 2 −11x c a n 2 −16n b y 2 +15y c a p 2 −22p b y 2 +5y c
p 2 − 1 3 p
q 2 + 3 4 q
m 2 + 2 5 m
a q 2 −6q b x 2 −7x c
n 2 − 2 3 n
u 2 +2u =3 z 2 +12z = −11 x 2 −20x =21
y 2 −2y =8 m 2 +4m = −44 n 2 −2n =−3
r 2 +6r = −11 t 2 −14t = −50 a 2 −10a =−5
b 2 +6b =41 x 2 +5x =2 y 2 −3y =2
u 2 −14u +12=−1 z 2 +2z −5=2 r 2 −4r −3=9
t 2 −10t −6=5 v 2 =9v +2 w 2 =5w −1
x 2 −5=10x y 2 −14=6y (x +6)(x −2) =9
⎛ ⎝y +9 ⎞ ⎠ ⎛ ⎝y +7 ⎞ ⎠ =80 (x +2)(x +4) =3 (x −2)(x −6) =5
3m 2 +30m −27=6 2x 2 −14x +12=0 2n 2 +4n =26
5x 2 +20x =15 2c 2 + c =6 3d 2 −4d =15
2x 2 +7x −15=0 3x 2 −14x +8=0 2p 2 +7p =14
3q 2 −5q =9 5x 2 −3x = −10 7x 2 +4x =−3

x 2 +10x = −25
a
y 2 +8y =48
b
c
a
b

2 − 4ab a = 3 b = −2.
108.
50.
x 2
⎛ ⎝ 1 2 · ba ⎞ ⎠
2
⎛
⎝ 1 2 b a ⎞ 2
⎠ = b 2
4a 2

− b 2a
a ≠0
x = −b ± b2 −4ac
2a
2x 2 +9x −5=0.

3y 2 − 5y + 2 = 0
4z 2 + 2z − 6 = 0

x 2 −6x =−5.
a, b, c.
a, b, c.
a 2 − 2a =15
b 2 + 24 = −10b
2x 2 +10x +11=0.

3m 2 + 12m + 7 = 0
5n 2 + 4n − 4 = 0
3p 2 +2p +9=0.
a, b, c.
a, b, c

a + bi
4a 2 − 2a + 8 = 0
5b 2 + 2b + 4 = 0
x(x +6) +4=0.
a, b, c.
a, b, c

x(x + 2) − 5 = 0.
3y ⎛
⎝y
− 2 ⎞ ⎠ − 3 = 0.
1 2 u2 + 2 3 u = 1 3 .

1
4
c 2 − 1 3 c = 1 12
1
9
d 2 − 1 2 d = − 1 3
4x 2 −20x = −25.
r 2 + 10r + 25 = 0.
25t 2 − 40t = −16.

2 −4ac
2x 2 +9x −5=0
9 2 −4·2(−5)
121
4x 2 −20x +25=0
(−20) 2 −4·4·25
0
3p 2 +2p +9=0
2 2 −4·3·9
−104
a ≠0,
a 3x 2 +7x −9=0 b 5n 2 + n +4=0 c 9y 2 −6y +1=0.
a

3x 2 +7x −9=0
a =3, b =7, c =−9
b 2 −4ac
(7) 2 −4·3·(−9)
49 + 108
157
b
5n 2 + n +4=0
a =5, b =1, c =4
b 2 −4ac
(1) 2 −4·5·4
1−80
−79
c
9y 2 −6y +1=0
a =9,b =−6,c =1
b 2 −4ac
(−6) 2 −4·9·1
36 − 36
0
a 8m 2 − 3m + 6 = 0 b 5z 2 + 6z − 2 = 0 c 9w 2 + 24w + 16 = 0.
a b 2 + 7b − 13 = 0 b 5a 2 − 6a + 10 = 0 c 4r 2 − 20r + 25 = 0.

ax 2 = k a(x − h) 2 = k
ax 2 = k a(x − h) 2 = k,
a 5z 2 =17b 4x 2 −12x +9=0 c 8u 2 +6u = 11.
a
5z 2 =17
ax 2 = k,
b
4x 2 −12x +9=0
c
8u 2 +6u =11
8u 2 +6u −11=0

a x 2 + 6x + 8 = 0 b (n − 3) 2 = 16 c 5p 2 − 6p = 9.
a 8a 2 + 3a − 9 = 0 b 4b 2 + 4b + 1 = 0 c 5c 2 = 125.

4m 2 + m −3=0 4n 2 −9n +5=0 2p 2 −7p +3=0
3q 2 +8q −3=0 p 2 +7p +12=0 q 2 +3q −18=0
r 2 −8r =33 t 2 +13t = −40 3u 2 +7u −2=0
2p 2 +8p +5=0 2a 2 −6a +3=0 5b 2 +2b −4=0
x 2 +8x −4=0 y 2 +4y −4=0 3y 2 +5y −2=0
6x 2 +2x −20=0 2x 2 +3x +3=0 2x 2 − x +1=0
8x 2 −6x +2=0 8x 2 −4x +1=0 (v +1)(v −5) −4=0
(x +1)(x −3) =2 ⎛ ⎝y +4 ⎞ ⎠ ⎛ ⎝y −7 ⎞ ⎠ =18 (x +2)(x +6) =21
1
3
m 2 + 1 12 m = 1 4
1
3
n 2 + n =− 1 2
3
4
b 2 + 1 2 b = 3 8
1
9
c 2 + 2 3 c =3 16c2 +24c +9=0 25d 2 −60d +36=0
25q 2 +30q +9=0 16y 2 +8y +1=0
a 4x 2 −5x +16=0 b
36y 2 +36y +9=0 c
a 9v 2 −15v +25=0
100w 2 +60w +9=0
b a r 2 +12r +36=0 b
c 8t 2 −11t +5=0 c
6m 2 +3m −5=0
5c 2 +7c −10=0
3v 2 −5v −1=0
a 25p 2 +10p +1=0 b
7q 2 −3q −6=0
7y 2 +2y +8=0
c

a x 2 −5x −24=0
b ⎛ ⎝y +5 ⎞ ⎠2 =12
c 14m 2 +3m =11
a (8v +3) 2 =81
b w 2 −9w −22=0
c 4n 2 −10n =6
a 6a 2 +14a =20
b ⎛ ⎝ x − 1 ⎞
4⎠2 = 5
16
c y 2 −2y =8
a 8b 2 +15b =4
b 5 9 v2 − 2 3 v =1
c ⎛ ⎝ w + 4 ⎞
3⎠2 = 2
9
x 2 +10x = 120
a
b
c
12y 2 +23y =24
a
b
c
a
b

y 4 − y 2 − 20.
⎛
⎝y
− 4 ⎞2 ⎠ + 8 ⎛
⎝y
− 4 ⎞ ⎠ + 15.
1
a x 2 1
· x
4
b ⎛ 1 ⎜⎛
⎞
⎝ x 3
⎟
⎠
2
c ⎛ −1⎞⎞ ⎝x
2 ⎠ .
u
⎛ ⎝x 2⎞ ⎠2
= x 4 .
u = x 2
x 2

6x 4 −7x 2 +2=0
x 4 − 6x 2 + 8 = 0
x 4 − 11x 2 + 28 = 0

(x −2) 2 +7(x −2) +12=0.
u = x −2
u x −2.
x.
(x − 5) 2 + 6(x − 5) + 8 = 0.
⎛
⎝y
− 4 ⎞2 ⎠ + 8 ⎛
⎝y
− 4 ⎞ ⎠ + 15 = 0.
( x) 2 = x.

x −3 x +2=0.
x ( x) 2 = x. u = x
u =
x
x.
x − 7 x + 12 = 0.
x − 6 x + 8 = 0.
2
x
3 1
−2x
3 −24=0.
1
⎛ 1
3
x ⎜x
⎝
3 ⎞ ⎠ ⎟ 2
2
= x
3 1
3
. u = x
1
3
u = x

1
x
3 .
x
2
x
3 1
− 5x
3 − 14 = 0.
1
x
2 1
− 8x
4 + 15 = 0.
3x −2 −7x −1 +2=0.
x −1 ⎛ ⎝x −1⎞ ⎠2
= x −2 . u = x −1
u = x −1
x −1 .
x x −1 = 1 x .

8x −2 − 10x −1 + 3 = 0.
6x −2 − 23x −1 + 20 = 0.

x 4 −7x 2 +12=0 x 4 −9x 2 +18=0 x 4 −13x 2 −30=0
x 4 +5x 2 −36=0 2x 4 −5x 2 +3=0 4x 4 −5x 2 +1=0
2x 4 −7x 2 +3=0 3x 4 −14x 2 +8=0
(x −3) 2 −5(x −3) −36=0
(x +2) 2 −3(x +2) −54=0
⎛ ⎝3y +2 ⎞2 ⎠ + ⎛ ⎝3y +2 ⎞ ⎠ −6=0
⎛
⎝5y −1 ⎞ ⎠2 +3 ⎛ ⎝5y −1 ⎞ ⎠ −28=0
⎛
⎝x 2 +1 ⎞ ⎠2
−5
⎛
⎝ x 2 +1 ⎞ ⎠ +4=0
⎛
⎝x 2 −4 ⎞ ⎠2
−4
⎛
⎝ x 2 −4 ⎞ ⎠ +3=0
2 ⎛ ⎝x 2 −5 ⎞ ⎠2
−5
⎛
⎝ x 2 −5 ⎞ ⎠ +2=0
2 ⎛ ⎝x 2 −5 ⎞ ⎠2
−7
⎛
⎝ x 2 −5 ⎞ ⎠ +6=0
x − x −20=0 x −8 x +15=0
x +6 x −16=0 x +4 x −21=0 6x + x −2=0
6x + x −1=0 10x −17 x +3=0 12x +5 x −3=0
2
x
3 1
+9x
3 2
+8=0 x
3 1
−3x
3 2
=28 x
3 1
+4x
3 =12
2
x
3 1
−11x
3 2
+30=0 6x
3 1
− x
3 2
=12 3x
3 1
−10x
3 =8
2
8x
3 1
−43x
3 2
+15=0 20x
3 1
−23x
3 1
+6=0 x −8x
2 +7=0
1
2x −7x
2 =15
6x −2 +13x −1 +5=0 15x −2 −26x −1 +8=0
8x −2 −2x −1 −3=0 15x −2 −4x −1 −4=0
a

2
x + 1 + 1
x − 1 = 1
x 2 − 1 .

Consecutive even integers
Consecutive odd integers
64, 66, 68 77, 79, 81
n 1 st even integer n 1 st odd integer
n +2 2 nd consecutive even integer n +2 2 nd consecutive odd integer
n +4 3 rd consecutive even integer n +4 3 rd consecutive odd integer

n =
n +2=
n(n +2) = 195
n 2 +2n = 195
n 2 +2n − 195 = 0
(n +15)(n −13) = 0
n +15=0 n −13=0
n = −15, n =13
13, 15 yes
−13, −15 yes
First odd integer n = 13 First odd integer n = −15
next odd integer n + 2 next odd integer n +2
13 + 2 −15 + 2
15 −13
13 · 15 = 195 yes
−13(−15) = 195 yes

A = 1 2 bh.
A = 1 2 bh
120 = 1 (2h +4)h
2
120 = h 2 +2h
h 2 +2h − 120 = 0
(h − 10)(h +12)=0
h −10=0 h +12=0
h = 10, h =−12
h = 10.
2h +4.
2·10+4
24

w =
3w −1=

a, b, c
a, b, c

x =
x =
a 2 + b 2 = c 2
x 2 + x 2 =10 2
2x 2 = 100
2x 2
2 = 100
2
x 2 =50
x =± 50
x =±52
x =5 2, x =−5 2

x ≈ 7.1
v 0 t
h = −16t 2 + v 0 t
t =
v 0
a, b, c.

a, b, c
t = D r .

=
D = r · t
t t = D r
2000
450 − r + 450 2000
+ r =9
(450 − r)(450 + r) ⎛ ⎝450 2000
− r + 450 2000 ⎞
+ r⎠ = 9(450 − r)(450 + r)
2000(450 + r) + 2000(450 − r) = 9(450 − r)(450 + r)
2000(450 + r + 450 − r) = 9 ⎛ ⎝450 2 − r 2⎞ ⎠
2000(900) = 9 ⎛ ⎝450 2 − r 2⎞ ⎠
2000(100) = 450 2 − r 2
200000 = 202500 − r 2
450 + 50 = 500 mph 2000
500 = 4 hours
450 − 50 = 400 mph 2000
400 = 5 hours
−2500 = − r 2
50 = r The speed of the jet stream.

x =
8x(x +12)

x =12

a
b
c
d
a
b
c
d
a
b

f (x) = x 2
2x 2 + 3x − 2 = 0.
− b 2a
f (x) =x 2
f (x) = ax 2 + bx + c a ≠0.
a ≠0,
f (x) = ax 2 + bx + c
f (x) =x 2

f (x) = x 2 −1.
f (x)
f (x) = x 2 −1
f (x) = −x 2 .
f (x) = x 2 + 1.

a f (x) =−3x 2 +2x −4 b f (x) =6x 2 +7x −9.
a
b

a f (x) = 2x 2 + 5x − 2 b f (x) = −3x 2 − 4x + 7.
a f (x) = −2x 2 − 2x − 3 b f (x) = 5x 2 − 2x − 1.
x =−
2a b .
x =−
2a b .

−
2a b .
x =−
2a b .
− b 2a .
x =− b 2a
f (x) =3x 2 −6x +2
a b
a
x =−
2a b
a, b

x =1
b
x =1
f (1)
(1, −1)
f (x) = 2x 2 − 8x + 1
a b
f (x) = 2x 2 − 4x − 3
a b

f (x) = ax 2 + bx + c
0 = ax 2 + bx + c
f (x) =0
(−1, 0) (−3, 0)

f (x) =0
a =−1,b =4,c =3
⎛ ⎝2+ 7,0 ⎞ ⎠
⎛
⎝2− 7,0 ⎞ ⎠
⎛
⎝2+ 7,0 ⎞ ⎠ ≈ (4.6, 0)
⎛
⎝2− 7,0 ⎞ ⎠ ≈ (−0.6, 0)

f (x) = ax 2 + bx + c :
y-intercept
x-intercepts
Let x = 0 and solve for f (x). Let f (x) = 0 and solve for x.
f (x) = x 2 −2x −8.
x =0
f (x)
x =0 f (0) =−8
(0, −8)
f (x) =0
x

f (x) =0 x =4orx =−2
(4, 0)
(−2, 0)
f (x) = x 2 + 2x − 8.
f (x) = x 2 − 4x − 12.
Quadratic equation
x 2 −2x −15 = 0
(x −5)(x +3) = 0
x −5=0 x +3 = 0
x =5 x = −3
Let f (x) =0.
Quadratic function
f (x) = x 2 −2x −15
0 = x 2 −2x −15
0 = (x −5)(x +3)
x −5 = 0 x +3=0
x = 5 x =−3
(5, 0) and (−3, 0)
x-intercepts
ax 2 + bx + c =0.

f (x) =5x 2 + x +4.
x =0
f (x)
x =0 f (0) =4
(0, 4)
f (x) =0
x
b 2 −4ac
1 2 −4·5·4
1−80
−79
f (x) = 3x 2 + 4x + 4.
f (x) = x 2 − 4x − 5.

−1
x =− b 2a x =3
x =3

f (3)
(3, 0).
x =0 f (0)
x =0

(0, −9).
(0, −9)
(6, −9)
(6, −9)
f (x) =0
f (x) =0

x =− b 2a
x =−2
x =−2.
f (x) x =−2.

(−2, 1)
x =0
f (0).
(0, 5)
(−4, 5)
(0, 5)
(−4, 5)

f (x) =0
f (x) =0

x =− b 2a
x =1
f (1)
x =1.
x =0
f (0)
1, −5 .
0, −3 .
(0, −3)
2, −3
(2, −3)
y =0
f (x) =0
a, b, c.

2.5, 0
−0.6, 0 .

f (x) = x 2 +2x −8.
x =−1
f (−1)
x =−1
−9
x =−1
(−1, −9)

f (x) = x 2 − 8x + 12.
f (x) = −4x 2 + 16x − 11.
h(t) = −16t 2 + v 0 t + h 0
a b
h(t) = −16t 2 + 176t +4
a

t = −
2a
b
t = −
2(−16)
176
t = 5.5
t = 5.5.
t = 5.5.
t = 5.5
b
h(5.5).
h(t) = −16t 2 + 176t +4
h(t) = −16(5.5) 2 + 176(5.5) +4
h(t) = 488
(5.5, 488).

f (x) =x 2 +3 f (x) = x 2 −3 y =−x 2 +1
f (x) =−x 2 −1
a f (x) =−2x 2 −6x −7 b a f (x) =4x 2 + x −4 b a f (x) =−3x 2 +5x −1 b
f (x) =6x 2 +2x +3
f (x) =−9x 2 −24x −16 f (x) =2x 2 −4x +5
a f (x) = x 2 +3x −4 b
f (x) =−4x 2 −12x −9
a b
f (x) = x 2 +8x −1 f (x) = x 2 +10x +25 f (x) =−x 2 +2x +5
f (x) =−2x 2 −8x −3
f (x) =x 2 +7x +6 f (x) =x 2 +10x −11 f (x) = x 2 +8x +12
f (x) = x 2 +5x +6 f (x) =−x 2 +8x −19 f (x) =−3x 2 + x −1
f (x) =x 2 +6x +13 f (x) = x 2 +8x +12 f (x) =4x 2 −20x +25
f (x) =−x 2 −14x −49 f (x) =−x 2 −6x −9 f (x) =4x 2 +4x +1
f (x) =x 2 +6x +5 f (x) =x 2 +4x −12 f (x) =x 2 +4x +3
f (x) =x 2 −6x +8 f (x) =9x 2 +12x +4 f (x) =−x 2 +8x −16
f (x) =−x 2 +2x −7 f (x) =5x 2 +2 f (x) =2x 2 −4x +1
f (x) =3x 2 −6x −1 f (x) =2x 2 −4x +2 f (x) =−4x 2 −6x −2

f (x) =−x 2 −4x +2 f (x) =x 2 +6x +8 f (x) =5x 2 −10x +8
f (x) = −16x 2 +24x −9 f (x) =3x 2 +18x +20 f (x) =−2x 2 +8x −10
f (x) =2x 2 + x −1 y =−4x 2 +12x −5 y = x 2 −6x +15
y =−x 2 +4x −5 y =−9x 2 +16 y =4x 2 −49
A(x) =x ⎛ ⎝ 120 − 2
x ⎞
⎠
A(x) =x ⎛ ⎝ 50 − 2
x ⎞
⎠
A(x) =x ⎛ ⎝ 150 − 2
x ⎞
⎠

f (x) = x 2
f (x) = x 2 −1
a
b

f (x) = x 2
y 2 − 14y + 49.
2x 2 − 16x + 32.
f(x) =x 2 + k
f (x) =x 2
f (x) =x 2
f (x) =x 2 + k.
f (x) = x 2 , g(x) = x 2 +2, h(x) = x 2 −2
f (x) =x 2

g(x) = x 2 +2 f (x) = x 2
h(x) = x 2 −2 f (x) = x 2
a f (x) = x 2 , g(x) = x 2 + 1, h(x) = x 2 − 1
b
a f (x) = x 2 , g(x) = x 2 + 6, h(x) = x 2 − 6
b
f (x) =x 2 + k,
f (x) =x 2 (k >0) (k

f (x) =x 2 −3
f (x) = x 2
k

f (x) = x 2
f (x) = x 2 − 5
f (x) = x 2 + 7
f(x) =(x − h) 2
f (x) =x 2
f (x) =x 2 + k.
f (x) =(x − h) 2 .
f (x) = x 2 , g(x) = (x −1) 2 , h(x) = (x +1) 2
f (x) =x 2

a f (x) = x 2 ,
g(x) = (x + 2) 2 , h(x) = (x − 2) 2
b
a f (x) = x 2 ,
g(x) = x 2 + 5, h(x) = x 2 − 5
b
f (x) =(x − h) 2 ,
f (x) =x 2
f(x) =(x − h) 2
f (x) =(x − h) 2 f (x) = x 2 h
|h|
f (x) =(x − h) 2 .
f (x) = x 2
f (x) =(x −6) 2

f (x) = x 2
f (x) = x 2
f (x) = (x − 4) 2
f (x) = (x + 6) 2
f (x) =(x − h) 2 + k

f (x) =(x +1) 2 −2
f (x) = x 2
f (x) = (x + 2) 2 − 3
f (x) = (x − 3) 2 + 1
f(x) =ax 2
f (x) =x 2
f (x) =ax 2 .

f (x) = x 2
f(x) =ax 2
f (x) =ax 2 f (x) = x 2
0

f (x) = −3x 2 .
f (x) = 2x 2 .
f (x) = x 2 + k, f (x) = (x − h) 2 , f (x) = ax 2
f (x) = ax 2 + bx + c
f (x) = a(x − h) 2 + k
f (x) =−3x 2 −6x −1 f (x) = a(x − h) 2 + k

x 2 −3
⎛ ⎝ 1 2 ⎠2 ·2⎞ =1
−3 −3
f (x) = a(x − h) 2 + k
f (x) =−4x 2 − 8x + 1 f (x) = a(x − h) 2 + k
f (x) = 2x 2 − 8x + 3 f (x) = a(x − h) 2 + k
f (x) = (x − h) 2 + k
f (x) = x 2 +6x +5
f (x) = a(x − h) 2 + k

⎛
⎝ 1 2 ·6⎞ ⎠2
=9
f (x) = (x − h) 2 + k
f (x) = x 2
f (x) = x 2
f (x) = x 2 + 2x − 3
f (x) = x 2 − 8x + 12
f (x) = a(x − h) 2 + k

f (x) =−2x 2 −4x +2
f (x) = a(x − h) 2 + k
x 2
−2
⎛
⎝ 1 2 ·2⎞ ⎠2
=1
−2 −2
f (x) = a(x − h) 2 + k
f (x) = x 2
f (x) = −3x 2 + 12x − 4
f (x) = −2x 2 + 12x − 9
f (x) = a(x − h) 2 + k

f (x) = a(x − h) 2 + k
f (x) = a(x − h) 2 + k
f (x) = a(x − h) 2 + k
a f (x) =2x 2 +4x +5 f (x) = a(x − h) 2 + k b
f (x) = a(x − h) 2 + k
f (x) =2x 2 +4x +5
f (x) =2(x 2 +2x)+5
f (x) =2(x 2 +2x +1)+5−2
f (x) =2(x +1) 2 +3
a, h, k. a =2 h =−1 k =3

a =2
x = h
(h, k)
f (0)
x =−1
(−1, 3)
f (0)=2⋅0 2 +4⋅0+5
f (0)=5
(0, 5)
(0, 5)
(−2, 5)
a f (x) = 3x 2 − 6x + 5 f (x) = a(x − h) 2 + k b
a f (x) = −2x 2 + 8x − 7 f (x) = a(x − h) 2 + k b

f (x) = a(x − h) 2 + k form.
(h, k), (−2, −1) h =−2 k = −1. f (x) = a(x − (−2)) 2 −1
a y (0, 7)
f (0) =7
a
7=a(0+2) 2 −1
7=4a −1
8=4a
2=a
h =−2,k =−1 a =2
f (x) = a(x − h) 2 + k
f (x) =2(x +2) 2 −1
f (x) = a(x − h) 2 + k

f(x) =x 2 + k
a b
f (x) = x 2 , g(x) = x 2 +4,
h(x) = x 2 −4.
f (x) = x 2 , g(x) = x 2 +7,
h(x) = x 2 −7.
f (x) =x 2 +3 f (x) =x 2 −7 g(x) = x 2 +2
g(x) = x 2 +5 h(x) = x 2 −4 h(x) = x 2 −5
f(x) =(x − h) 2
a b
h
f (x) = x 2 , g(x) = (x −3) 2 ,
h(x) = (x +3) 2 .
f (x) = x 2 , g(x) = (x +4) 2 ,
h(x) = (x −4) 2 .
f (x) =(x −2) 2 f (x) =(x −1) 2 f (x) =(x +5) 2
f (x) =(x +3) 2 f (x) =(x −5) 2 f (x) =(x +2) 2
f (x) =(x +2) 2 +1 f (x) =(x +4) 2 +2 f (x) =(x −1) 2 +5
f (x) =(x −3) 2 +4 f (x) =(x +3) 2 −1 f (x) =(x +5) 2 −2
f (x) =(x −4) 2 −3
f (x) =(x −6) 2 −2
f(x) =ax 2
f (x) =−2x 2 f (x) =4x 2 f (x) =−4x 2
f (x) =−x 2 f (x) = 1 2 x2 f (x) = 1 3 x2
f (x) = 1 4 x2
f (x) = − 1 2 x2

f (x) =a(x−h) 2 +k
f (x) =−3x 2 −12x −5 f (x) =2x 2 −12x +7 f (x) =3x 2 +6x −1
f (x) =−4x 2 −16x −9
a f (x) =a(x−h) 2 +k b
f (x) = x 2 +6x +5 f (x) = x 2 +4x −12 f (x) =x 2 +4x +3
f (x) = x 2 −6x +8 f (x) = x 2 −6x +15 f (x) = x 2 +8x +10
f (x) =−x 2 +8x −16 f (x) =−x 2 +2x −7 f (x) =−x 2 −4x +2
f (x) =−x 2 +4x −5 f (x) =5x 2 −10x +8 f (x) =3x 2 +18x +20
f (x) =2x 2 −4x +1 f (x) =3x 2 −6x −1 f (x) =−2x 2 +8x −10
f (x) =−3x 2 +6x +1
a f (x) =a(x−h) 2 +k b
f (x) =2x 2 +4x +6 f (x) =3x 2 −12x +7 f (x) =−x 2 +2x −4
f (x) =−2x 2 −4x −5
a f (x) =x 2 +4 b f (x) =x 2 −4 c
f (x) = (x+4) 2 d f (x) = (x−4) 2 e f (x) = (x+4) 2 −4 f f (x) = (x+4) 2 +4 g f (x) = (x−4) 2 −4 h
f (x) = (x−4) 2 +4

f (x) =a(x−h) 2 +k
f (x) = x 2 +4x +5
f (x) =2x 2 −4x −3

a
b

2x − 3 = 0.
2y 2 + y =15
1
x 2 + 2x − 8 > 0
ax 2 + bx + c 0
ax 2 + bx + c ≥0

x 2 −6x +8

a x 2 + 2x − 8 < 0 b
a x 2 − 8x + 12 ≥ 0 b
f (x) = ax 2 + bx + c.
−x 2 −8x −12≤0
f (x) =−x 2 −8x −12
−x 2 −8x −12≤0
x =−
2a
b
x =−
2(−1)
−8
x =−4
f (x) =−x 2 −8x −12
f (−4) = −(−4) 2 − 8(−4) − 12
f (−4) = −16 + 32 − 12
f (−4) = 4
(−4, 4)

f (x) =0
f (x) =−x 2 −8x −12
0=−x 2 −8x −12
0 = −1(x +6)(x +2)
x =−6 x =−2
(−6, 0), (−2, 0)
(−∞, −6] ∪ [−2, ∞)
a −x 2 − 6x − 5 > 0 b
a −x 2 + 10x − 16 ≤ 0 b
x 2 − x −12≥0

x 2 + 2x − 8 ≥ 0
x 2 − 2x − 15 ≤ 0
x 2 − x −12

x 2 +6x −7≥0
−1
a, b, c
−x 2 +6x −7≥0
x 2 −6x +7≤0
x 2 −6x +7=0
x = −b ± b2 −4ac
2a
x = −(−6) ± (−6)2 −4⋅1⋅(7)
2⋅1
x = 6± 8
2
x = 6±2 2
2
x = 2⎛ ⎝3± 2 ⎞ ⎠
2
x =3± 2
x =3+ 2 x =3− 2
x ≈ 1.6 x ≈ 4.4
−x 2 +6x −7≥0
⎡
⎣3− 2, 3+ 2 ⎤ ⎦
−x 2 + 2x + 1 ≥ 0
−x 2 + 8x − 14 < 0

a ≠0.
a x 2 −3x +4>0 b x 2 −3x +4≤0
a
−x 2 −3x +4>0
a, b, c
x 2 −3x +4=0
x = −b ± b2 −4ac
2a
x = −(−3) ± (−3)2 −4⋅1⋅(4)
2⋅1
x =
x =
3± −7
2
3± 7i
2

x 2 −3x +4>0. x
(−∞, ∞).
b
x 2 −3x +4≤0
x 2 −3x +4=0
x 2 −3x +4≤0.
a −x 2 + 2x − 4 ≤ 0 b −x 2 + 2x − 4 ≥ 0
a x 2 + 3x + 3 < 0 b x 2 + 3x + 3 > 0

a b
x 2 +6x +5>0 x 2 +4x −12−15 x 2 +8x 0
x 2 −3x +6≤0 −x 2 + x −7>0 −x 2 −4x −5

x = −b ± b2 −4ac
,
2a
a ≠0,
f (x) = ax 2 + bx + c.
x 2 = k x = k or x =− k x =± k
⎛ ⎝ 1 2 ⎠2 b⎞ ,
⎛ ⎝ 1 2
2 b⎞ ⎠
⎛ ⎝ 1 2 · ⎠2 b⎞ ,

a ≠0
x = −b ± b2 −4ac
2a
a ≠0,

A = 1 2 bh.
f (x) = ax 2 + bx + c,
f (x) = ax 2 + bx + c
x =− b 2a .
− b 2a .
x =− b 2a
f (x) = ax 2 + bx + c :

y-intercept
x-intercepts
Let x = 0 and solve for f (x). Let f (x) = 0 and solve for x.
f (x) =x 2 + k
f (x) =x 2 + k f (x) = x 2
|k|
f (x) =(x − h) 2
f (x) =(x − h) 2 f (x) = x 2
|h|
f (x) =ax 2
f (x) =ax 2 f (x) = x 2
0

f (x) = ax 2 + bx + c
y 2 = 144 n 2 −80=0 4a 2 = 100
2b 2 =72 r 2 +32=0 t 2 +18=0
2
3
w 2 −20=30 5c 2 +3=19
a(x − h) 2 = k
⎛
⎝p −5 ⎞ ⎠2 +3=19 (u +1) 2 =45
⎛
⎝ x − 1 ⎞
4⎠2 = 3
16
⎛
⎝ y − 2 2
⎞
3⎠
= 2 (n −4) 2 − 50 = 150 (4c −1) 2 = −18
9
n 2 +10n +25=12 64a 2 +48a +9=81
x 2 +22x m 2 −8m a 2 −3a
b 2 +13b
d 2 +14d = −13 y 2 −6y =36 m 2 +6m = −109
t 2 −12t = −40 v 2 −14v = −31 w 2 −20w = 100

m 2 +10m − 4 = −13 n 2 −6n +11=34 a 2 =3a +8
b 2 =11b −5 (u +8)(u +4) =14 (z −10)(z +2) =28
3p 2 −18p +15=15 5q 2 +70q +20=0 4y 2 −6y =4
2x 2 +2x =4 3c 2 +2c =9 4d 2 −2d =8
2x 2 +6x =−5 2x 2 +4x =−5
4x 2 −5x +1=0 7y 2 +4y −3=0 r 2 − r −42=0
t 2 +13t +22=0 4v 2 + v −5=0 2w 2 +9w +2=0
3m 2 +8m +2=0 5n 2 +2n −1=0 6a 2 −5a +2=0
4b 2 − b +8=0 u(u −10) +3=0 5z(z −2)=3
1
8
p 2 − 1 5 p =−1 20
2
5
q 2 + 3 10 q = 1 10
4c 2 +4c +1=0
9d 2 −12d =−4
a 9x 2 −6x +1=0
b 3y 2 −8y +1=0
c 7m 2 +12m +4=0
d 5n 2 − n +1=0
a 5x 2 −7x −8=0
b 7x 2 −10x +5=0
c 25x 2 −90x +81=0
d 15x 2 −8x +4=0
a 16r 2 −8r +1=0
b 5t 2 −8t +3=9
c 3(c +2) 2 =15
a 4d 2 +10d −5=21
b 25x 2 −60x +36=0
c 6(5v −7) 2 = 150

x 4 −14x 2 +24=0 x 4 +4x 2 −32=0 4x 4 −5x 2 +1=0
⎛
⎝2y +3 ⎞ ⎠2 +3 ⎛ ⎝2y +3 ⎞ ⎠ −28=0
x +3 x −28=0 6x +5 x −6=0
2
x
3 1
−10x
3 1
+24=0 x +7x
2 +6=0
8x −2 −2x −1 −3=0

y = x 2 −2 y =−x 2 +3
a y =−3x 2 +3x −1
b y =5x 2 +6x +3
a y = x 2 +8x −1
b y =−4x 2 −7x +1
a b
y =−x 2 +6x +8 y =2x 2 −8x +1
y = x 2 −4x −5 y = x 2 −8x +15 y = x 2 −4x +10
y =−5x 2 −30x −46 y =16x 2 −8x +1 y = x 2 +16x +64
y = x 2 +8x +15 y = x 2 −2x −3 y =−x 2 +8x −16
y =4x 2 −4x +1 y = x 2 +6x +13 y =−2x 2 −8x −12
y =7x 2 +14x +6 y =−3x 2 +12x −10

A(x) =x ⎛ ⎝ 90 − 2
x ⎞
⎠
f(x) =x 2 + k
g(x) = x 2 +4 h(x) = x 2 −3
f (x) =(x +1) 2 g(x) =(x −3) 2
f (x) =(x +2) 2 +3 f (x) =(x +3) 2 −2 f (x) =(x −1) 2 +4
f (x) =(x −4) 2 −3
f(x) =ax 2
f (x) =2x 2 f (x) =−x 2 f (x) = 1 2 x2
f (x) =a(x−h) 2 +k
f (x) =2x 2 −4x −4 f (x) =3x 2 +12x +8
a f (x) =a(x−h) 2 +k b
f (x) =3x 2 −6x −1 f (x) =−2x 2 −12x −5 f (x) =2x 2 +4x +6

f (x) =3x 2 −12x +7
a f (x) =a(x−h) 2 +k b
f (x) =−3x 2 −12x −5 f (x) =2x 2 −12x +7
f (x) =a(x−h) 2 +k
x 2 − x −6>0 x 2 +4x +3≤0 −x 2 − x +2≥0
−x 2 +2x +30 −x 2 + x −6>0 x 2 −2x +4≥0

3(w +5) 2 = 27.
a 2 −8a + 7 = 23.
2m 2 −5m +3=0.
2x(3x −2) −1=0 9y 2 −3y +1=0
4
6p 2 −13p +7=0 3q 2 −10q +12=0
4x 4 −17x 2 +4=0
2
y
3 1
+2y
3 −3=0
a b c d
y =3x 2 +6x +8 y =−x 2 −8x −16
f (x) = x 2 +6x +9 f (x) =−2x 2 +8x +4
f (x) =(x +3) 2 +2 f (x) = x 2 −4x −1
x 2 −6x −8≤0 2x 2 + x −10>0

f (x) = 2x − 3 g(x) = x 2 + 2x − 3, f (4).
x, 3x + 2y = 12.

(x + 4)
5 − 4.
5
f g, f ∘g
⎛ ⎝f ∘g ⎞ ⎠(x) = f ⎛ ⎝g(x) ⎞ ⎠.
f ⎛ ⎝g(x) ⎞ ⎠ “ f g x.”
g(x),
f ∘g
f ⎛ ⎝g(x) ⎞ ⎠ f g
⎛
⎝f ∘g ⎞ ⎠(x) = f ⎛ ⎝g(x) ⎞ ⎠
g(x) = x 2
⎛ ⎝f ∘g ⎞ ⎠(x) = f ⎛ ⎝g(x) ⎞ ⎠ f (x) = x −4.
⎛ ⎝f ∘g ⎞ ⎠(x),
⎛
⎝g∘ f ⎞ ⎠(x) ⎛ ⎝f · g ⎞ ⎠(x)
f (x) =4x −5 g(x) =2x +3, a ⎛ ⎝f ∘g ⎞ ⎠(x), b ⎛ ⎝g∘ f ⎞ ⎠(x), c ⎛ ⎝f · g ⎞ ⎠(x).
a

⎛ ⎝f ∘g ⎞ ⎠(x).
b
⎛ ⎝f ∘g ⎞ ⎠(x).
a b
c ⎛ ⎝f · g ⎞ ⎠(x) ⎛ ⎝f ∘g ⎞ ⎠(x). a c
Use the definition of ⎛ ⎝f · g ⎞ ⎠(x).
⎛
⎝f · g ⎞ ⎠(x) = f (x) · g(x)
Substitute f (x) =4x − 5 and g(x) =2x +3.
⎛
⎝f · g ⎞ ⎠(x) = (4x −5) · (2x +3)
Multiply.
⎛
⎝f · g ⎞ ⎠(x) =8x 2 +2x −15
f (x) = 3x − 2 g(x) = 5x + 1, a ⎛
⎝f
∘g ⎞ ⎠(x) b ⎛ ⎝g∘ f ⎞ ⎠(x) c ⎛
⎝f
· g ⎞ ⎠ ⎞ (x)
f (x) = 4x − 3, g(x) = 6x − 5, a ⎛
⎝f
∘g ⎞ ⎠(x), b ⎛ ⎝g∘ f ⎞ ⎠(x), c ⎛
⎝f
· g ⎞ ⎠ ⎞ (x).
f (x) = x 2 −4, g(x) =3x +2, a ⎛ ⎝f ∘g ⎞ ⎠(−3), b ⎛ ⎝g∘ f ⎞ ⎠(−1), c ⎛ ⎝f ∘ f ⎞ ⎠(2).
a

⎛ ⎝f ∘g ⎞ ⎠(−3).
b
⎛ ⎝g∘ f ⎞ ⎠(−1).
c
⎛ ⎝f ∘ f ⎞ ⎠(2).
f (x) = x 2 − 9, g(x) = 2x + 5, a ⎛
⎝f
∘g ⎞ ⎠(−2), b ⎛ ⎝g∘ f ⎞ ⎠(−3), c ⎛
⎝f
⎛
∘ f ⎞ ⎠ ⎞ (4).
f (x) = x 2 + 1, g(x) = 3x − 5, a ⎛
⎝f
∘g ⎞ ⎠(−1), b ⎛ ⎝g∘ f ⎞ ⎠(2), c ⎛
⎝f
∘ f ⎞ ⎠ ⎞ (−1).

a {(−3, 27), (−2, 8), (−1, 1), (0, 0), (1, 1), (2, 8), (3, 27)} b {(0, 0), (1, 1), (4, 2), (9, 3), (16, 4)}.
a
{(−3, 27), (−2, 8), (−1, 1), (0, 0), (1, 1), (2, 8), (3, 27)}
(−3, 27) (3, 27),
b
{(0, 0), (1, 1), (4, 2), (9, 3), (16, 4)}
a {(−3, −6), (−2, −4), (−1, −2), (0, 0), (1, 2), (2, 4), (3, 6)} b
{(−4, 8), (−2, 4), (−1, 2), (0, 0), (1, 2), (2, 4), (4, 8)}
a {(27, −3), (8, −2), (1, −1), (0, 0), (1, 1), (8, 2), (27, 3)} b
{(7, −3), (−5, −4), (8, 0), (0, 0), (−6, 4), (−2, 2), (−1, 3)}

a b
a
b

f {(0, 5), (1, 6), (2, 7), (3, 8)}. x
f y y
x f f −1 .
f f −1 x y f
f −1 f −1 f .
f (x) (x, y), f −1 (x)
(y, x).
{(0, 3), (1, 5), (2, 7), (3, 9)}.

x y
x y
{(0, 3), (1, 5), (2, 7), (3, 9)}
{(3, 0), (5, 1), (7, 2), (9, 3)}
{3, 5, 7, 9}
{0, 1, 2, 3}
{(0, 4), (1, 7), (2, 10), (3, 13)}.
{(−1, 4), (−2, 1), (−3, 0), (−4, 2)}.
f (x) (x, y),
f −1 (x) (y, x).
(a, b) f (x), (b, a) f −1 (x).
(a, b) (b, a) y = x.
y = x.
f (x) f −1 (x),
y = x.

(−5, −3), (−3, −1), (−1, 0), (0, 2), (3, 4)
(−3, −5), (−1, −3), (0, −1), (2, 0), (4, 3)
y = x.

f −1 ⎛ ⎝f (x) ⎞ ⎠ = x, for all x in the domain of f
f ⎛ ⎝f −1 (x) ⎞ ⎠ = x, for all x in the domain of f −1
f (x) =5x −1 g(x) = x +1
5
g ⎛ ⎝f (x) ⎞ ⎠ = x f ⎛ ⎝g(x) ⎞ ⎠ = x.

5x −1 f (x).
x +1
5
g(x).
g ⎛ ⎝f (x) ⎞ ⎠ = x f ⎛ ⎝g(x) ⎞ ⎠ = x f (x) =5x −1 g(x) = x +1
5
f (x) = 4x −3 g(x) = x + 3
4 .
f (x) = 2x + 6 g(x) = x − 6
2 .
f (x) (x, y),
f −1 (x) (y, x).
f (x) =4x +7.

f (x) = 5x − 3.
f (x) = 8x + 5.
f (x).
f −1 (x)
f (x) =
5
2x −3.

f (x) =
5
2x −3
y f (x)
5
y = 2x −3
x y
5
x = 2y −3
y
(x) 5 = ⎛ 5
5 ⎞
⎝ 2y −3 ⎠
x 5 =2y −3
x 5 +3=2y
x 5 +3
2
= y
f −1 (x) y f −1 (x) = x5 +3
2
f −1 ⎛ ⎝f (x) ⎞ ⎠ = ? x
f ⎛ ⎝f −1 (x) ⎞ ⎠ = ? x
f −1 ⎛5 ⎞
⎝ 2x −3 ⎠ = ? x f ⎛ ⎝ x5 +3⎞
2 ⎠ =? x
5
⎛5 ⎞
⎝ 2x −3 ⎠ +3
= ? x
2
2x −3+3
2
= ? x
2x
2
= ? x
x = x ✓
5
2 ⎛ ⎝ x5 +3⎞
2 ⎠ −3 = ? x
5
x 5 +3−3
5
x 5
= ? x
= ? x
x = x ✓
5
f (x) = 3x − 2.
4
f (x) = 6x − 7.

a b c
f (x) =4x +3 g(x) =2x +5
f (x) =3x −1 g(x) =5x −3
f (x) =6x −5 g(x) =4x +1
f (x) =2x +7 g(x) =3x −4
f (x) =3x g(x) =2x 2 −3x
f (x) =2x g(x) =3x 2 −1
f (x) =2x −1 g(x) = x 2 +2
f (x) =4x +3 g(x) = x 2 −4
f (x) =2x 2 +3 g(x) =5x −1,
a ⎛ ⎝f ∘g ⎞ ⎠(−2) b ⎛ ⎝g∘ f ⎞ ⎠(−3) c ⎛ ⎝f ∘ f ⎞ ⎠(−1)
f (x) =5x 2 −1 g(x) =4x −1,
a ⎛ ⎝f ∘g ⎞ ⎠(1) b ⎛ ⎝g∘ f ⎞ ⎠(−1) c ⎛ ⎝f ∘ f ⎞ ⎠(2)
f (x) =2x 3 g(x) =3x 2 +2,
a ⎛ ⎝f ∘g ⎞ ⎠(−1) b ⎛ ⎝g∘ f ⎞ ⎠(1) c (g∘g)(1)
f (x) =3x 3 +1
g(x) =2x 2 −3,
(g∘g)(1)
a ⎛ ⎝f ∘g ⎞ ⎠(−2) b ⎛ ⎝g∘ f ⎞ ⎠(−1) c
{(−3, 9), (−2, 4), (−1, 1), (0, 0)
(1, 1), (2, 4), (3, 9)}
{(9, −3), (4, −2), (1, −1), (0, 0)
(1, 1), (4, 2), (9, 3)}
{(−3, −5), (−2, −3), (−1, −1)
(0, 1), (1, 3), (2, 5), (3, 7)}
{(5, 3), (4, 2), (3, 1), (2, 0)
(1, −1), (0, −2), (−1, −3)}

a
a
b
b
a
a
b
b
{(2, 1), (4, 2), (6, 3), (8, 4)} {(6, 2), (9, 5), (12, 8), (15, 11)}
{(0, −2), (1, 3), (2, 7), (3, 12)} {(0, 0), (1, 1), (2, 4), (3, 9)}
{(−2, −3), (−1, −1), (0, 1), (1, 3)} {(5, 3), (4, 2), (3, 1), (2, 0)}

f (x) = x +8 g(x) = x −8 f (x) = x −9 g(x) = x +9
f (x) =7x g(x) = x 7
f (x) =7x +3 g(x) = x −3
7
f (x) = x
11
g(x) =11x
f (x) =5x −4 g(x) = x −4
5
f (x) = x +2 g(x) = x 2 3
−2 f (x) = x −4
g(x) = x 3 +4
f (x) = x −12 f (x) = x +17
f (x) =9x f (x) =8x
f (x) = x 6
f (x) = x 4
f (x) =6x −7
f (x) =−2x +5
f (x) =7x −1
f (x) =−5x −4
f (x) = x 2 +6, x ≥0 f (x) = x 2 −9, x ≥0
f (x) = x 3 −4 f (x) = x 3 +6

f (x) = 1
x +2
f (x) = 1
x −6
f (x) = x −2, x ≥2 f (x) = x +8, x ≥−8
f (x) =
f (x) =
3
x −3
4
9x −5, x ≥ 5 9
3
f (x) = x +5
f (x) =
4
8x −3, x ≥ 3 8
5
f (x) = −3x +5
f (x) =
5
−4x −3
a
b

⎛ x
3⎞ ⎝ ⎠ .
x 2
⎛ ⎞ 0 ⎠ .
a 2 0 b ⎛ 1
⎝ 3
a 2 −1 b ⎛ 1
⎝ 3
⎛ ⎞ −1 ⎠ .
f (x) = a x a >0 a ≠1.
a >0 a ≠1,
f (x) = a x
a ≠1. a =1, f (x) = a x f (x) =1 x . 1 x =1
f (x) =1.
a >0. −4, f (x) = (−4) x
x = 1 2 .

f (x) = (−4) x
f ⎛ ⎝ 1 1
⎞
2⎠ = (−4) 2
f ⎛ ⎝ 1 ⎞
2⎠ = −4 not a real number
f (x) = (−4) x x
a >0.
f (x) =2 x g(x) =3 x .
f (x) = 4 x .
g(x) = 5 x .
f (x) =2 x g(x) =3 x , f (x) =4 x g(x) =5 x ,
a >1.
(0, 1). a 0 =1
f (x) = a x (1, a). f (x) =2 x (1, 2)

g(x) =3 x (1, 3). a 1 = a.
f (x) = a x ⎛ ⎝ −1, 1 a ⎞ ⎠ . f (x) =2x
⎛
⎝ −1, 1 ⎞
2⎠ g(x) =3x ⎛ ⎝ −1, 1 ⎞
3⎠ . a−1 = 1 a .
(−∞, ∞).
x
(0, ∞).
x y =0,
f(x) = a x a >1
(−∞, ∞)
(0, ∞)
(0, 1)
(1, a),
⎛
⎝ −1, 1 a ⎞ ⎠
x y =0
f (x) = a x a >0,
a >1. 0

f (x) = ⎛1⎞ x
⎝ 4⎠
.
g(x) = ⎛1⎞ x
⎝ 5⎠
.
0

(−∞, ∞). x
(0, ∞).
a >1.
f(x) = a x
a >1 0

f (x) = 2 x g(x) = 2 x − 1 .
f (x) = 3 x g(x) = 3 x + 1 .
f (x) =2 x g(x) =2 x +1
f (x) =3 x g(x) =3 x −2.

f (x) = 3 x g(x) = 3 x + 2.
f (x) = 4 x g(x) = 4 x − 2.
f (x) =3 x g(x) =3 x −2
⎛ ⎝ 1+1 n ⎞ n
⎠
⎛ ⎝ 1+1 n ⎞ n
⎠ n.
n
1+1 n n
e ≈ 2.718281827

π
e
⎛ ⎝ 1+1 n ⎞ ⎠n
,
e ≈ 2.718281827...
e,
f (x) = e x
e
f (x) = e x
(−∞, ∞) (0, ∞).
f (x) = e x g(x) =2 x h(x) =3 x .
20 a ≠1,
If a x = a y , then x = y.
3 2x −5 = 27.

3 3x − 2 = 81.
7 x − 3 = 7.
ex2
e 3 = e2x
e x2
e 3 = e2x
am
a n = am − n .
e x2 −3 = e 2x
x 2 −3=2x
x 2 −2x −3=0

(x −3)(x +1) =0
x =3, x =−1
ex2
e x = e2 .
ex2
e x = e6 .
A = P ⎛ ⎝1+ r n ⎞ ⎠nt .
A = Pe rt .
A = P ⎛ ⎝1+ n r ⎞ nt
⎠
A = Pe rt
when compounded n times a year.
when compounded continuously.
$10,000 5%,
a
b
c

A =?
P = $10,000
r = 0.05
t = 18 years
a
n =4
A = P ⎛ ⎝1+ n r ⎞ nt
⎠
A = 10,000 ⎛ ⎝ 1+0.05 4
4·18
⎞
⎠
A = $24,459.20
b
n =12
A = P ⎛ ⎝1+ n r ⎞ nt
⎠
A = 10,000 ⎛ ⎝ 1+0.05 12
A = $24,550.08
12 · 18
⎞
⎠
c
A = Pe rt
0.05 · 18
A = 10,000e
A = $24,596.03
$15,000 4%,
a
b
c

5%,
a
b
c
A = Pe rt
A 0 ,
P,
A 0 ,
A = A 0 e rt
A =?
A 0 = 100
r = 0.25/hour
t = 24 hours
A = A 0 e rt 0.25 · 24
A = 100e
A = 40,342.88
A = 40,343
15%

10%

f (x) =2 x
f (x) =6 x
f (x) = (1.5) x
f (x) = ⎛ ⎝ 1 x
⎞
2⎠
f (x) = ⎛ ⎝ 1 x
⎞
6⎠
f (x) = (0.4) x
g(x) =3 x
g(x) =7 x
g(x) = (2.5) x
g(x) = ⎛ ⎝ 1 x
⎞
3⎠
g(x) = ⎛ ⎝ 1 x
⎞
7⎠
g(x) = (0.6) x
f (x) =4 x , g(x) =4 x −1
f (x) =3 x , g(x) =3 x −1
f (x) =2 x , g(x) =2 x −2 f (x) =2 x , g(x) =2 x +2
f (x) =3 x , g(x) =3 x +2 f (x) =4 x , g(x) =4 x +2
f (x) =2 x , g(x) =2 x +1
f (x) =2 x , g(x) =2 x −1
f (x) =3 x +2
f (x) =3 x −2
f (x) =2 x +3
f (x) = ⎛ ⎝ 1 x −4
⎞
2⎠
f (x) = e x +1
f (x) =−2 x
f (x) =2 x −3
f (x) = ⎛ ⎝ 1 ⎞
2⎠x −3
f (x) = e x −2
f (x) =2 −x −1 −1
2 3x −8 =16 2 2x −3 =32
3 x +3 =9 3 x2 =81

4 x2 =4
4 x =32
4 x +2 =64 4 x +3 =16
2 x2 +2x = 1 2
3 x2 −2x = 1 3
e 3x · e 4 = e 10 e 2x · e 3 = e 9
e x2
e 2 = ex
e x2
e 3 = e2x
a 2 x b 2 x+1 c 2 x−1 d 2 x +2 e 2 x −2
f 3 x

$5,000
20%
a b
c
$12,000
6%,
a
b c
$5,000
8%
6%
1.12%
$8,000
6%
20%
0.9%
y = x 2
y =2 x
x
x
a
b

x 2 = 81.
3 −2 .
2 4 = 3x − 5.
f (x) = a x
Rewrite with y = f (x).
y = a x
Interchange the variables x and y. x = a y
Solve for y. Oops! We have no way to solve for y!
f (x) = a x .
f −1 (x) = log a x
f (x) = log a x a a >0, x >0, a ≠1.
y = log a x is equivalent to x = a y
y = log a x x = a y

a 2 3 1
=8, b 5
2 = 5, c ⎛ ⎝ 1 2⎞
⎠x
= 1 . 16
a 3 2 1
= 9 b 7
2 = 7 c ⎛1⎞ x
⎝ 3⎠
=
27
1
a 4 3 1
= 64 b 4
3 3
= 4
c ⎛1⎞ x
⎝ 2⎠
=
32
1
a 2 = log 8 64, b 0 = log 4 1, c −3 = log 10
1
1000 .
a 3 = log 4 64 b 0 = log x 1 c −2 = log 10
1
100
a 3 = log 3 27 b 0 = log x 1 c −1 = log 10
1
10

a log x 36 = 2, b log 4 x =3, c log 12
1
8
= x.
a
log x 36 = 2
x 2 =36
x =6, x =−6
x =−6
x = 6 Therefore, log 6 36 = 2.
b
log 4 x =3
4 3 = x
x = 64 Therefore, log 4 64 = 3.
c
log 12
1
8
= x
1 8 ⎛ ⎝ 1 3
⎞
2⎠
⎛
⎝ 1 ⎞
2⎠x = 1
8
⎛
⎝ 1 x
⎞
2⎠
3
⎛ = 1⎞ ⎝2⎠ x = 3 Therefore, log 12
1
8
=3
x : a log x 64 = 2 b log 5 x = 3 c log 12
1
4
= x
x : a log x 81 = 2 b log 3 x = 5 c log 13
1
27
= x
log 3 27,
27”? 3 3 = 27,
“3
log 3 27 = 3. x

a log 5 25, b log 9 3, c log 2
1
16
.
a
log 5 25
log 5 25 = 2
x
log 5 25 = x
5 x =25
5 2 5 x =5 2
x = 2 Therefore, log 5 25 = 2.
b
log 9 3
x log 9 3 = x
9 x =3
3 2
⎛
⎝3 2⎞ x
⎠ =3
1
3 2x =3 1
2x =1
x = 1 2
Therefore, log 9 3= 1 2 .
c

log 2
1
16
x
log 2
1
16
= x
2 x = 1 16
2 4 2 x = 1 2 4
2 x =2 −4
x = −4 Therefore, log 2
1
16
=−4.
a log 12 144 b log 4 2 c log 2
1
32
a log 9 81 b log 8 2 c log 3
1
9
y = log a x, x = a y .
y = log 2 x.
y = log 2 x, 2 y = x.

y 2 y = x x, y
−2 2 −2 = 1 2 2 = 1 4
−1 2 −1 = 1 2 1 = 1 2
⎛
⎝ 1 4 ,2⎞ ⎠
⎛
⎝ 1 2 ,−1⎞ ⎠
2 0 =1 (1, 0)
2 1 =2 (2, 1)
2 2 =4 (4, 2)
2 3 =8 (8, 3)
y = log 3 x.
y = log 5 x.
y = log 2 x, y = log 3 x, y = log 5 x
a >1.
(1, 0). 0 = log a 1
a 0 =1
(a, 1). 1 = log a a a 1 = a.
y = log a x ⎛ ⎝ 1 a ,−1⎞ ⎠ . −1 = log a 1 a
a −1 = 1 a ,

(0, ∞).
(−∞, ∞).
x =0,
y = log a x a >1
(0, ∞)
(−∞, ∞)
x-intercept (1, 0)
y-intercept
(a, 1),
⎛
⎝ 1 a ,−1⎞ ⎠
y-axis
y = log a x 0

y
1
3y = x x, y
−2 ⎛
⎝
1
3
⎞
⎠
−2
=3 2 =9 (9, −2)
−1 ⎛
⎝
1
3
⎞
⎠
−1
=3 1 =3 (3, −1)
⎛
⎝ 1 0
⎞
3⎠
=1 (1, 0)
⎛
⎝ 1 ⎞
3⎠1 = 1 ⎛1 3 ⎝ ,1 ⎞ 3 ⎠
⎛
⎝ 1 ⎞
3⎠2 = 1 ⎛1 9 ⎝ ,2 ⎞ 9 ⎠
⎛
⎝ 1 3
⎞
3⎠
=
27
1
⎛
⎝ 1 27 ,3⎞ ⎠
y = log 12 x.
y = log 14 x.
y = log 12 x, y = log 13 x y = log 14 x
0

y = log a x
a >1 0

y = x.
f (x) = log e x f (x) =lnx x.”
f (x) =lnx e, x >0.
y =lnx is equivalent to x = e y
f (x) = log x 10 x >0.
y = log x is equivalent to x =10 y
y = log a x
a >0 a ≠1. x >0.
a log a 49 = 2 b ln x =3.
a

log a 49 = 2
a 2 =49
a = ±7
a =−7. a =7, a =−7
a = 7 log a 49 = 2
log 7 49 = ? 2
7 2 = ? 49
49 = 49 ✓
b
ln x =3
e 3 = x
x = e 3 ln x = 3
ln e 3 = ? 3
e 3 = e 3 ✓
a log a 121 = 2 b ln x = 7
a log a 64 = 3 b ln x = 9
a log 2 (3x −5) =4 b ln e 2x =4.
a

log 2 (3x −5) =4
2 4 =3x −5
16 = 3x −5
21 = 3x
7=x
x = 7 log 2 (3x −5) = 4
log 2 (3⋅7−5) = ? 4
log 2 (16) = ? 4
2 4 = ? 16
16 = 16 ✓
b
ln e 2x =4
e 4 = e 2x
4=2x
2=x
x =2 lne 2x = 4
ln e 2·2 = ? 4
ln e 4 = ? 4
e 4 = e 4 ✓
a log 2 (5x − 1) = 6 b ln e 3x = 6
a log 3 (4x + 3) = 3 b ln e 4x = 4
10 −12
D = 10 log ⎛ ⎝
I ⎞
10 −12 ⎠

10 −2
log10 10 = 10.
10 −7
10 −3
R R = log I,
I
R = log I,

R = log I
1906 earthquake 7.8 = log I
Convert to exponential form. I =10 7.8
2014 earthquake 5.1 = log I
Convert to exponential form. I =10 5.1
Intensity for 1906
Intensity for 2014
10 7.8
10 5.1
10 2.7
501

4 2 =16 2 5 =32
3 3 =27 5 3 = 125
10 3 = 1000 10 −2 = 1
100
1
x
2 1
= 3 x
3 3
= 6
32 x 4
= 32
17 x 5
= 17
⎛ ⎝ 1 ⎞
4⎠2 = 1 ⎛ 16 ⎝ 1 ⎞
3⎠4 = 1
81
3 −2 = 1 9
4 −3 = 1 64
e x =6
e 3 = x
3 = log 4 64 6 = log 2 64
4 = log x 81 5 = log x 32
0 = log 12 1 0 = log 7 1
1 = log 3 3 1 = log 9 9
−4 = log 10
1
10,000
3 = log 10 1,000
5 = log e x x = log e 43
x
log x 49 = 2 log x 121 = 2
log x 27 = 3 log x 64 = 3
log 3 x =4 log 5 x =3
log 2 x =−6 log 3 x =−5

log 14
1
16
= x
log 13
1
9
= x
log 14 64 = x
log 19 81 = x
log 7 49 log 6 36
log 4 1 log 5 1
log 16 4 log 27 3
log 12 2 log 12 4
log 2
1
16
log 3
1
27
log 4
1
16
log 9
1
81
y = log 2 x
y = log 4 x
y = log 6 x
y = log 7 x
y = log 1.5 x
y = log 2.5 x
y = log 13 x
y = log 15 x
y = log 0.4 x
y = log 0.6 x
log a 16 = 2 log a 81 = 2
log a 8=3 log a 27 = 3
log a 32 = 2 log a 24 = 3
ln x =5 ln x =4
log 2 (5x +1) =4 log 2 (6x +2) =5
log 3 (4x −3) =2 log 3 (5x −4) =4
log 4 (5x +6) =3 log 4 (3x −2) =2

ln e 4x =8 ln e 2x =6
logx 2 =2
log(x 2 −25)=2
log 2
⎛
⎝ x 2 −4 ⎞ ⎠ =5 log 3
⎛
⎝ x 2 +2 ⎞ ⎠ =3
10 −6
10 −10
10 −2
8.2
7.0
10 −2
6.7
5.1
log a a x = x. log 7 32
a
b

a a 0 b a 1 .
3
x 2 y.
a 0 =1,
log a 1=0. a 1 = a, log a a =1.
log a 1 = 0 log a a =1
a log 8 1 b log 6 6.
a
log 8 1
log a 1=0 0 log 8 1=0
b
log 6 6
Use the property, log a a = 1. 1 log 6 6=1

a log 13 1 b log 9 9.
a log 5 1 b log 7 7.
a log a x = x loga x = log a x,
log a a x = x a x = a x ,
a >0, x >0 a ≠1,
a log a x = x
loga a x = x
a 4 log 4 9 b log 3 3 5 .
a
4 log 4 9
a log a x = x 9 4
log 4 9 =9
b
log 3 3 5
a log a x = x 5 log 3 3 5 =5
a 5 log 5 15 b log 7 7 4 .
a 2 log 2 8 b log 2 2 15 .
a m · a n = a m + n ,
log a M · N = log a M + log a N

M >0,N >0,a>0 a≠1,
log a (M · N) = log a M + log a N
a log 3 7x
b log 4 64xy.
a
log 3 7x
log a (M · N) = log a M + log a N
log 3 7 + log 3 x
log 3 7x = log 3 7 + log 3 x
b
log 4 64xy
log a (M · N) = log a M + log a N
log 4 64
log 4 64 + log 4 x + log 4 y
3 + log 4 x + log 4 y
log 4 64xy = 3 + log 4 x + log 4 y
a log 3 3x b log 2 8xyx
a log 9 9x b log 3 27xyx
a m
a n = am − n ,
log M a = log N a M − log a N
M >0,N >0,a>0 a≠1,

log a
M
N
= log a M − log a N
log a M − log a N ≠ log a (M − N).
a log 5
5
7
b log x
100
a
log 5
5
7
log a
M
N
= log a M − log a N log 5 5 − log 5 7
1 − log 5 7
log 5
5
7
= 1 − log 5 7
b
log x
100
log a
M
N
= log a M − log a N
log x − log100
log x −2
log
100 x = log x −2
a log 3 4 4
b log
1000
x
a log 2
5
4
b log 10 y
(a m ) n = a m · n ,
log a M p = p log a M

M >0, a>0, a≠1 p
log a M p = p log a M
a log 5 4 3 b logx 10
a
log 5 4 3
log a M p = p log a M 3log 5 4
log 5 4 3 = 3log 5 4
b
logx 10
log a M p = p log a M
10 log x
logx 10 = 10log x
a log 7 5 4 b logx 100
a log 2 3 7 b logx 20
M >0, N >0,a>0, a≠1 p

a e
log a 1=0 ln 1 = 0
log a a =1 ln e =1
a log a x = x
log a a x = x
e ln x = x
ln e x = x
log a (M · N) = log a M + log a N ln(M · N) =lnM +lnN
log a
M
N
= log a M − log a N ln M N
=lnM −lnN
log a M p = p log a M ln M p = p ln M
log 4
⎛
⎝ 2x 3 y 2⎞ ⎠
log 4
⎛
⎝ 2x 3 y 2⎞ ⎠
log a M · N = log a M + log a N log 4 2 + log 4 x 3 + log 4 y 2
log a M p = p log a M
log 4 2 + 3log 4 x + 2log 4 y
1
2 + 3log 4 x + 2log 4 y
log 4
⎛
⎝ 2x 3 y 2⎞ ⎠ = 1 2 + 3log 4 x + 2log 4 y
log 2
⎛
⎝ ⎛ 5x 4 y 2⎞ ⎠ ⎞
log 3
⎛
⎝ ⎛ 7x 5 y 3⎞ ⎠ ⎞

4
log x 3
2
3y 2
z
4
log x 3
2
3y 2 z
log a M p = p log a M
log a M · N = log a M − log a N
log a M · N = log a M + log a N
log a M p = p log a M
1
4
⎛
log 2 ⎜ x3 ⎞
⎝3y 2 ⎟
z⎠
⎛
1log 4 2 ⎜ x3 ⎞
⎝3y 2 ⎟
z⎠
1⎛
⎛
4⎝log 2 ⎝ x 3⎞ ⎛
⎠ − log 2 ⎝ 3y 2 z ⎞⎞ ⎠⎠
1⎛
⎛
4⎝log 2 ⎝ x 3⎞ ⎠ − ⎛ ⎝log 2 3 + log 2 y 2 + log 2 z ⎞⎞ ⎠⎠
1⎛
4
⎝3log 2 x − ⎛ ⎝log 2 3 + 2log 2 y + log 2 z ⎞ ⎠ ⎞ ⎠
1⎛
4
⎝3log 2 x − log 2 3 − 2log 2 y − log 2 z ⎞ ⎠
4
log x 3
2
3y 2 z
= 1 ⎛
4
⎝3log 2 x − log 2 3 − 2log 2 y − log 2 z ⎞ ⎠
5
log x 4
4
2y 3 z 2
3
log x 2
3
5y z
log 4 3 + log 4 x − log 4 y

log a M + log a N = log a M · N
log a M − log a N = log M a N
log 4 3 + log 4 x − log 4 y
log 4 3x − log 4 y
log 4
3x y
log 4 3 + log 4 x − log 4 y = log 4
3x y
log 2 5 +log 2 x − log 2 y
log 3 6 − log 3 x − log 3 y
2log 3 x + 4log 3 (x +1)
2log 3 x + 4log 3 (x +1)
log a M + log a N = log a M · N log 3 x 2 + log 3 (x +1) 4
log a M + log a N = log a M · N
log 3 x 2 (x +1) 4
2log 3 x + 4log 3 (x +1) = log 3 x 2 (x +1) 4
3log 2 x + 2log 2 (x − 1)
2log x + 2log(x + 1)

log a M
y = log a M
log a M
y = log a M
a y = M
log b log b a y = log b M
y
y = log a M
ylog b a = log b M
y = log b M
log b a
log a M = log b M
log b a
b. b >0,b ≠1.
a, b M >0,
log a M = log b M
log b a
log a M = logM
loga
log a M = ln M
ln a
new base b new base 10 new base e
(≈)
log 4 35.
log35
log4
log 3 42.
log 5 46.

a log 4 1 b log 8 8
a log 12 1 b ln e
a 3 log 3 6 b log 2 2 7 a 5 log 5 10 b log 4 4 10
a 8 log 8 7 b log 6 6 −2
a 6 log 6 15 b log 8 8 −4
a 10 log 5 b log10 −2
a 10 log 3 b log10 −1
a e ln4 b ln e 2 a e ln3 b ln e 7
log 4 6x
log 5 8y
log 2 32xy
log 3 81xy
log100x
log1000y
log 3
3
8
log 6
5
6
log 4
16 y log 5
125 x
log x
10 log 10,000
y
ln e3
3
ln e4
16
log 3 x 2 log 2 x 5
logx −2
logx −3
log 4 x 3
log 5 x
ln x 3 3
ln x 4

log 5
⎛
⎝ 4x 6 y 4⎞ ⎠
log 2
⎛
⎝ 3x 5 y 3⎞ ⎠
⎛
log 3 ⎝ 2x 2⎞ ⎛4
⎠ log 5 ⎝ 21y 3 ⎞
⎠
xy
log 2
3
z 2 log 4ab 3 c 4
5
d 2
log x
3
4
16y 4 log x 2
3
27y 4
log 2
2x + y 2
3x +2y
z 2 log 2
3
5z 2
4
log 5x 3
2
2y 2 z 4
3
log 3x 2
5
4y 3 z
log 6 4 + log 6 9
log4 + log25
log 2 80 − log 2 5 log 3 36 − log 3 4
log 3 4 + log 3 (x +1) log 2 5 − log 2 (x −1)
log 7 3 + log 7 x − log 7 y
4log 2 x + 6log 2 y
log 5 2 − log 5 x − log 5 y
6log 3 x + 9log 3 y
log 3
⎛
⎝ x 2 −1 ⎞ ⎠ − 2log 3 (x −1) log ⎛ ⎝x 2 +2x +1 ⎞ ⎠ − 2log(x +1)
4log x − 2logy − 3logz
3ln x + 4ln y − 2ln z
1log x − 3log(x +1) 2log(2x +3) + 1 log(x +1)
3 2
log 3 42 log 5 46
log 12 87 log 15 93
log 2 17 log 3 21

log a 5x,
log a (5+x).
log(a + b) ≠ loga + logb.
log a x p ⎛
, ⎝log a x ⎞ ⎠r .
log 7 15
a
b 1 − 10,

x 2 = 16.
x 2 − 5x + 6 = 0.
x(x + 6) = 2x + 5.
log a M = log a N M = N.
M >0,N > 0, a>0, a≠1
If log a M = log a N, then M = N.
2log 5 x = log 5 81.

2 log 5 x = log 5 81
log 5 x 2 = log 5 81
log a M = log a N M = N x 2 =81
x =±9
x =−9 x =9, x =−9
x = 9 2log 5 x = log 5 81
2log 5 9 = ? log 5 81
log 5 9 2 = ? log 5 81
log 5 81 = log 5 81 ✓
2log 3 x = log 3 36
3log x = log64
log 3 x + log 3 (x −8) =2.

log 3 x + log 3 (x −8) =2
log a M + log a N = log a M ⋅ N log 3 x(x −8) =2
3 2 = x(x −8)
9=x 2 −8x
0=x 2 −8x −9
0=(x −9)(x +1)
x −9 = 0, x +1=0
x =9, x =−1
x = −1 log 3 x + log 3 (x −8) =2
log 3 (−1) + log 3 (−1−8) = ? 2
x = 9 log 3 x + log 3 (x −8) =2
log 3 9 + log 3 (9−8) = ? 2
2+0 = ? 2
2=2✓
log 2 x + log 2 (x − 2) = 3
log 2 x + log 2 (x − 6) = 4
log 4 (x +6) − log 4 (2x +5) = −log 4 x.

log 4 (x +6) − log 4 (2x +5) = −log 4 x
⎛
log x +6⎞
4 ⎝ 2x +5⎠ = log 4 x−1
x −1 = 1 x log ⎛
4 ⎝2x x +6 ⎞
+5⎠ = log 1 4 x
log a M = log a N
M = N
x +6
2x +5 = 1 x
x(x +6) =2x +5
x 2 +6x =2x +5
x 2 +4x −5=0
(x +5)(x −1) =0
x +5=0, x −1=0
x =−5, x =1
log(x + 2) − log(4x + 3) = −log x.
log(x − 2) − log(4x + 16) = log 1 x .
5 x = 11.

5 x = 11
x
x.
log5 x = log11
xlog5 = log11
x = log11
log5
x ≈ 1.490
5 1 =5 5 2 = 25, 5 1.490 ≈11?
7 x = 43.
8 x = 98.
3e x +2 = 24.
3e x +2 =24
e x +2 =8
ln e x +2 =ln8
x
ln e =1
(x +2)ln e =ln8
x +2 = ln8
x =ln8−2
x ≈ 0.079
2e x − 2 = 18.
5e 2x = 25.

A = P ⎛ ⎝1+ n r ⎞ nt
⎠
A = Pe rt
when compounded n times a year.
when compounded continuously.
A = $50,000
P = $10,000
r =?
t = 17 years
A = Pe rt
50,000 = 10,000e r ·17
r.
5=e 17r
ln5 = ln e 17r
ln5 = 17rln e
ln5 = 17r
ln5
17 = r
r ≈ 0.095
r ≈ 9.5%
9.5%
$10,000 $150,000
$15,000 $90,000

A = A 0 e kt . k
A 0 ,
A = A 0 e kt
A = 300
A 0 = 100
k = ?
t = 3 hours
A = A 0 e kt
300 = 100e k ·3
k
3=e 3k
ln3 = ln e 3k
ln3 = 3kln e
ln3 = 3k
ln3
3
= k
k ≈ 0.366
A = ?
A 0 = 100
k = ln3
3
t = 24 hours
A = A 0 e kt
ln3
3
A = 100e
·24
A ≈ 656,100

A = 50
A 0 = 100
k = ?
t = 1590years
A = A 0 e kt
50 = 100e k · 1590
k
0.5 = e 1590k
ln0.5 = ln e 1590k
ln0.5 = 1590kln e
ln0.5 = 1590k
ln0.5 = k exact answer
1590
A = ?
A 0 = 100
k = ln0.5
1590
t = 500years
A = A 0 e kt
ln0.5
1590
A = 100e
· 500
A ≈ 80.4 mg

log 4 64 = 2log 4 x
log49 = 2log x
3log 3 x = log 3 27 3log 6 x = log 6 64
log 5 (4x −2) = log 5 10
log 3
⎛
⎝ x 2 +3 ⎞ ⎠ = log 3 4x
log 3 x + log 3 x =2 log 4 x + log 4 x =3
log 2 x + log 2 (x −3) =2 log 3 x + log 3 (x +6) =3
log x + log(x +3) =1 log x + log(x −15) =2
log(x +4) − log(5x +12) = −log x
log(x −1) − log(x +3) = log 1 x
log 5 (x +3) + log 5 (x −6) = log 5 10 log 5 (x +1) + log 5 (x −5) = log 5 7
log 3 (2x −1) = log 3 (x +3) + log 3 3
log(5x +1) = log(x +3) + log2
3 x =89 2 x =74
5 x = 110 4 x = 112
e x =16 e x =8
⎛ ⎝ 1 x
⎞
⎛
2⎠x =6 1⎞ ⎝3⎠ =8
4e x +1 =16 3e x +2 =9
6e 2x =24 2e 3x =32
1
4
e x =3 1
3
e x =2
e x +1 +2=16 e x −1 +4=12
3 3x +1 =81 6 4x −17 = 216

e x2 = e5x
14
e
e x2
e x = e20
log a 64 = 2 log a 81 = 4
ln x =−8 ln x =9
log 5 (3x −8)=2 log 4 (7x + 15) = 3
ln e 5x =30 ln e 6x =18
3log x = log125 7log 3 x = log 3 128
log 6 x + log 6 (x − 5) = log 6 24 log 9 x + log 9 (x − 4) = log 9 12
log 2 (x +2) − log 2 (2x +9)=−log 2 x
log 6 (x +1) − log 6 (4x + 10) = log 6
1 x
6 x =91
⎛ ⎝ 1 ⎞
2⎠x =10
7e x −3 =35 8e x +5 =56
$5,000
$10,000
$15,000
$30,000
$5,000
7%.
⎛ ⎞
⎝A =2A 0⎠ .
$8,000
5%.
⎛ ⎞
⎝A =2A 0⎠ .

3 x +1 = 81, 3 x +1 = 75.
a
b

f (x) = log x 10,
x >0.
y = log x is equivalent to x =10 y
a >0 a ≠1, f (x) = a x .
a ≠1.
f (x) = log a x a, a >0,
y = log a x is equivalent to x = a y
x >0,
⎛ ⎝ 1+1 n ⎞ ⎠n
,
e ≈ 2.718281827...
f (x) = e x .
(−∞, ∞) (0, ∞).
f (x) =lnx e, x >0.
y =lnx is equivalent to x = e y
f g, f ∘g
⎛
⎝f ∘g ⎞ ⎠(x) = f ⎛ ⎝g(x) ⎞ ⎠
f ⎛ ⎝g(x) ⎞ ⎠ f g x.
f (x)
(x, y), f −1 (x) (y, x).
x f f −1 ,
f −1 ⎛ ⎝f (x) ⎞ ⎠ = x
f ⎛ ⎝f −1 (x) ⎞ ⎠ = x
f (x).
f −1 (x) y.

f (x) = a x :
a >1 0

a >1 00 a ≠1
log a 1 = 0 log a a =1
a log a x = x loga a x = x
M >0,N >0,a>0 a≠1,
log a M · N = log a M + log a N

M >0,N >0,a>0 a≠1,
log a
M
N
= log a M − log a N
M >0, a>0, a≠1 p
log a M p = p log a M
M >0, a>0, a≠1 p
a e
log a 1=0 ln1 = 0
log a a =1 ln e =1
a log a x = x
log a a x = x
e ln x = x
ln e x = x
log a (M · N) = log a M + log a N ln(M · N) =lnM +lnN
log a
M
N
= log a M − log a N ln M N
=lnM −lnN
log a M p = p log a M ln M p = pln M
M >0,
log a M = log b M
log b a
log a M = logM
loga
log a M = ln M
ln a
new base b new base 10 new base e
M >0,N >0, a >0, a≠1
If log a M = log a N, then M = N.
A = P ⎛ ⎝1+ n r ⎞ nt
⎠
A = Pe rt
when compounded n times a year.
when compounded continuously.
A 0
A = A 0 e rt .

a b c
f (x) =7x −2
g(x) =5x +1
f (x) =4x
g(x) = x 2 +3x
f (x) =3x 2 +2
g(x) =4x −3,
a ⎛ ⎝f ∘g ⎞ ⎠(−3)
b ⎛ ⎝g∘ f ⎞ ⎠(−2)
c ⎛ ⎝f ∘ f ⎞ ⎠(−1)
f (x) =2x 3 +5
g(x) =3x 2 −7,
a ⎛ ⎝f ∘g ⎞ ⎠(−1)
b ⎛ ⎝g∘ f ⎞ ⎠(−2)
c (g∘g)(1)
{(−3, −5), (−2, −4), (−1, −3), (0, −2), {(−3, 0), (−2, −2), (−1, 0), (0, 1),
(−1, −1), (−2, 0), (−3, 1)}
(1, 2), (2, 1), (3, −1)}
{(−3, 3), (−2, 1), (−1, −1), (0, −3),
(1, −5), (2, −4), (3, −2)}

a
a
b
b
{(−3, 10), (−2, 5), (−1, 2), (0, 1)}
f (x) =3x +7
g(x) = x −7
3
f (x) =2x +9
g(x) = x +9
2
f (x) =6x −11 f (x) = x 3 +13

f (x) = 1
x +5
5
f (x) = x −1
f (x) =4 x f (x) = ⎛ ⎝ 1 x
5⎞
⎠
g(x) = (0.75) x g(x) =3 x +2
f (x) = (2.3) x −3 f (x) = e x +5
f (x) =−e x
3 5x −6 =81 2 x2 =16
9 x =27
5 x2 +2x = 1 5
e 4x · e 7 = e 19 e x2 = e2x
15
e
$12,000
4%
a
b
c
$20,000
7%
15%
0.7%
5 4 = 625 10 −3 =
1,000
1
1
63
5 5
= 63
e y =16

7 = log 2 128 5 = log100,000
4=lnx
log x 125 = 3 log 7 x =−2
log 12
1
16
= x
log 2 32 log 8 1
log 3
1
9
y = log 5 x y = log 14 x
y = log 0.8 x
log a 36 = 5 ln x =−3
log 2 (5x −7) =3 ln e 3x =24
log(x 2 − 21) = 2
10 −3
a log 7 1 b log 12 12 a 5 log 5 13 b log 3 3 −9
a 10 log 5 b log10 −3 a e ln8 b ln e 5

⎛
log 4 ⎝64xy ⎞ ⎠ log10,000m
log 49 7 y ln e 5
2
logx −9 7
log 4 z
⎛
log 3 ⎝ 4x 7 y 8⎞ ⎠ log 8a 2 b 6 c 5
d 3
ln 3x2 − y 2
3
z 4 log 7x 2
6
6y 3 z 5
log 2 56 − log 2 7 3log 3 x + 7log 3 y
log 5
⎛
⎝ x 2 −16 ⎞ ⎠ − 2log 5 (x +4) 1
4
logy − 2log ⎛ ⎝y −3 ⎞ ⎠
log 5 97 log 3 16
3log 5 x = log 5 216 log 2 x + log 2 (x −2) =3
log(x −1) − log(3x +5) = −log x log 4 (x −2) + log 4 (x + 5) = log 4 8
ln(3x −2) =ln(x +4) +ln2
2 x = 101 e x =23
⎛
⎝ 1 x
⎞
3⎠
=7 7e x +3 =28
e x −4 +8=23

$18,000
$30,000
$4500
6%.
⎛
⎞
⎝A =2A 0⎠ .

f (x) =6x +1
g(x) =8x −3, a ⎛ ⎝f ∘g ⎞ ⎠(x), b ⎛ ⎝g∘ f ⎞ ⎠(x),
c ⎛ ⎝f · g ⎞ ⎠(x).
{(−2, 2), (−1, −3), (0, 1), (1, −2), (2, −3)}
a
b
f (x) = x 5 −9. g(x) =2 x −3 .
2 2x −4 = 64.
ex2
e 4 = e3x .
$21,000
5%,
a
b
c
3 = log 7 343
10 −2 = 1
100 .
log 5 x =−3
log 11 1. log 4
1
64
.

y = log 3 x.
log(x 2 −39)=1
10 −8
a 6 log 6 17
b log 9 9 −3
log 5 25ab
ln e12
8
4
log 5x 3
2
16y 2 z 7
5log 4 x + 3log 4 y 1log x − 3log(x +5)
6
log 4 73.
log 7 (x +2) + log 7 (x − 3) = log 7 24
⎛
⎝ 1 x
⎞
5⎠
=9
5e x −4 =40
4%.
⎛ ⎞
⎝A =2A 0⎠ .

x 2 − 18x + 81.

x 2 − 12x − 12 = 0.
(6, 4) (2, 1).

a 2 + b 2 = c 2
3 2 +4 2 = d 2
9 + 16 = d 2
25 = d 2
d =5 d =−5
d =−5.
(6, 4)
(2, 1)
(6, 1) (2, −2).
(5, 3) (−3, −3).
(x 1 , y 1 )
(x 2 , y 2 ).
6−2 x 2 − x 1 .
4−1 y 2 − y 1 .
x 1 − x 2 y 1 − y 2 . x 2 − x 1
x 1 − x 2
| x 2 − x 1| | y 2 − y 1|.
| x 2 − x 1| | y 2 − y 1|

a 2 + b 2 = c 2
⎛
⎝|x 2 − x 1| ⎞2 ⎠ + ⎛ ⎝|y 2 − y 1| ⎞2 ⎠ = d 2
(x 2 − x 1 ) 2 + (y 2 − y 1 ) 2 = d 2
d =± (x 2 − x 1 ) 2 + (y 2 − y 1 ) 2
d = (x 2 − x 1 ) 2 + (y 2 − y 1 ) 2
(x 1 , y 1 ) (x 2 , y 2 ).
(x 1 , y 1 ) (x 2 , y 2 )
d = (x 2 − x 1 ) 2 + (y 2 − y 1 ) 2
(−5, −3) (7, 2).
d = (x 2 − x 1 ) 2 + (y 2 − y 1 ) 2
⎛ x 1 , y 1 ⎞
−5, −3
⎝ ⎠
, ⎛ x 2 , y 2
⎝ 7, 2 ⎞
⎠
d = ⎛ ⎝7−(−5) ⎞2 ⎠ + (2−(−3)) 2
d = 12 2 +5 2
d = 144 + 25
d = 169
d =13
(−4, −5) (5, 7).
(−2, −5) (−14, −10).
(10, −4) (−1, 5).

d = (x 2 − x 1 ) 2 + (y 2 − y 1 ) 2
⎛ x 1 , y 1 ⎞
10, −4
⎝ ⎠
, ⎛ x 2 , y 2
⎝ −1, 5 ⎞
⎠
d = (−1 − 10) 2 + ⎛ ⎝5−(−4) ⎞ 2
⎠
d = (−11) 2 +9 2
d = 121 + 81
d = 202
d = 202
or
d ≈ 14.2
(−4, −5) (3, 4).
(−2, −5) (−3, −4).
(x 1 , y 1 ) (x 2 , y 2 )
⎛
⎝
x 1 + x 2
2
, y 1 + y 2⎞
2 ⎠
(−5, −4) (7, 2).
⎛
⎝
x 1 + x 2
2
, y 1 + y 2⎞
2 ⎠

⎛ x 1 , y 1 ⎞
−5, −4
⎝ ⎠
, ⎛ x 2 , y 2
⎝ 7, 2 ⎞
⎠
⎛ −5 + 7 ⎝ ,
2
−4 2
+ 2 ⎞
⎠
⎛2 ⎝ , −2 ⎞
2 2 ⎠
(1, −1)
(1, −1).
(−3, −5) (5, 7).
(−2, −5) (6, −1).
(x 1 , y 1 ) (x 2 , y 2 ).

(h, k),
(h, k),
(h, k),
(x, y).
(h, k), (x, y)
d = (x 2 − x 1 ) 2 + (y 2 − y 1 ) 2
r =
(x − h) 2 + ⎛ ⎝y − k ⎞ 2
⎠
r 2 = (x − h) 2 + ⎛ ⎝y − k ⎞ ⎠2
(h, k),
(h, k),

(0, 0).
(x − h) 2 + ⎛ ⎝y − k ⎞ ⎠2 = r 2
r =3,h =0, k =0. (x −0) 2 + ⎛ ⎝y −0 ⎞ ⎠2 =3 2
x 2 + y 2 =9
(0, 0).
(0, 0).
(0, 0). (0, 0),
x 2 + y 2 = r 2 .
(−1, 3).
(x − h) 2 + ⎛ ⎝y − k ⎞2 ⎠ = r 2
(x − (−1)) 2 + ⎛ ⎝y −3 ⎞ ⎠2 =2 2
(x +1) 2 + ⎛ ⎝y −3 ⎞ ⎠2 =4
(2, −4).
(−3, −5).
(2, 4) (−2, 1).

(2, 4) (−2, 1)
r = (x 2 − x 1 ) 2 + (y 2 − y 1 ) 2
⎛ x 1 , y 1
⎝ 2, 4 ⎞
⎠
, ⎛ x 2 , y 2
⎝ −2, 1 ⎞
⎠ r = (−2 − 2) 2 + (1−4) 2
r = (−4) 2 + (−3) 2
r = 16+9
r = 25
r =5
r =5, (2, 4),
(x − h) 2 + ⎛ ⎝y − k ⎞2 ⎠ = r 2
(x −2) 2 + ⎛ ⎝y −4 ⎞2 ⎠ =5 2
(x −2) 2 + ⎛ ⎝y −4 ⎞ ⎠2 =25
(2, 1) (−2, −2).
(7, 1) (−1, −5).

(x − h) 2 + ⎛ ⎝y − k ⎞ ⎠2 = r 2 (h, k),
x +2,
(x +2) 2 + ⎛ ⎝y −1 ⎞ ⎠2 =9.
(h, k)
(−2, 1)
a b (x − 3) 2 + ⎛
⎝y
+ 4 ⎞2 ⎠ = 4.
a b (x − 3) 2 + ⎛
⎝y
− 1 ⎞2 ⎠ = 16.
x 2 , y 2
4x 2 +4y 2 = 64.

(h, k)
(0, 0)
a b 3x 2 + 3y 2 = 27
a b 5x 2 + 5y 2 = 125
(x +2) 2 + ⎛ ⎝y −1 ⎞ ⎠2 =9,
(x +2) 2 + ⎛ ⎝y −1 ⎞ ⎠2 =9
x 2 +4x +4+y 2 −2y +1=9
x 2 + y 2 +4x −2y −4=0
x 2 + y 2 + ax + by + c =0
a b x 2 + y 2 −4x −6y +4=0.

(2, 3)
a b x 2 + y 2 − 6x − 8y + 9 = 0.
a b x 2 + y 2 + 6x − 2y + 1 = 0.
y 2 x 2
a b x 2 + y 2 +8y =0.
y 2 +8y.
(0, −4)

a b x 2 + y 2 − 2x − 3 = 0.
a b x 2 + y 2 − 12y + 11 = 0.

(2, 0) (5, 4) (−4, −3) (2, 5) (−4, −3) (8, 2)
(−7, −3) (8, 5) (−1, 4) (2, 0) (−1, 3) (5, −5)
(1, −4) (6, 8) (−8, −2) (7, 6) (−3, −5) (0, 1)
(−1, −2) (−3, 4) (3, −1) (1, 7) (−4, −5) (7, 4)
a b
(0, −5) (4, −3) (−2, −6) (6, −2)
(3, −1) (4, −2) (−3, −3) (6, −1)
(0, 0).
2 5
(3, 5) (−2, 6)
2.5, (1.5, −3.5)
1.5, (−5.5, −6.5)
(3, −2) (3, 6) (6, −6) (2, −3)
(4, 4) (2, 2) (−5, 6) (−2, 3)
a b
(x +5) 2 + ⎛ ⎝y +3 ⎞ ⎠2 =1 (x −2) 2 + ⎛ ⎝y −3 ⎞ ⎠2 =9 (x −4) 2 + ⎛ ⎝y +2 ⎞ ⎠2 =16
(x +2) 2 + ⎛ ⎝y −5 ⎞ ⎠2 =4 x 2 + ⎛ ⎝y +2 ⎞ ⎠2 =25 (x −1) 2 + y 2 =36
(x − 1.5) 2 + ⎛ ⎝y + 2.5 ⎞ ⎠2 = 0.25
(x −1) 2 + ⎛ ⎝y −3 ⎞2 ⎠ = 9 4
x 2 + y 2 =64

x 2 + y 2 =49 2x 2 +2y 2 =8 6x 2 +6y 2 = 216
a b
x 2 + y 2 +2x +6y +9=0 x 2 + y 2 −6x −8y =0 x 2 + y 2 −4x +10y −7=0
x 2 + y 2 +12x −14y +21=0
x 2 + y 2 +6y +5=0 x 2 + y 2 −10y =0
x 2 + y 2 +4x =0 x 2 + y 2 −14x +13=0
a
b

y = −3x 2 + 12x − 12.
x 2 − 6x + 6 = 0.
y = 3x 2 − 6x + 5.

y = ax 2 + bx + c
y = ax − h 2 + k
a >0 a 0 a

−1,
x =− b 2a . x =3.
x =3.
x =3.
(3, 1).

x =0.
x =0.
(0, −8).
(0, −8)
(6, −8).
(6, −8).
y =0.
y =0.
(4, 0), (2, 0).

y = −x 2 + 5x − 6
y = −x 2 + 8x − 12
y = a(x − h) 2 + k.
y =3x 2 −6x +5
y = a(x − h) 2 + k
y =3x 2 −6x +5
y =3 ⎛ ⎝x 2 −2x
⎞
⎠ +5
y =3 ⎛ ⎝x 2 −2x +1 ⎞ ⎠ +5−3
y =3(x −1) 2 +2
a =3 h =1 k =2
a =2,
x = h. x =1.
(h, k). (1, 2).

x =0. y =3(x −1) 2 +2
y =3·0 2 −6·0+5
y =5
(0, 5)
(0, 5)
(2, 5)
y = 3(x −1) 2 +2
0 = 3(x −1) 2 +2
−2 = 3(x −1) 2
− 2 3
= (x −1)2
± − 2 3 = x −1
a y = 2x 2 + 4x + 5 b
a y = −2x 2 + 8x − 7 b

x = ay 2 + by + c
x = a y − k 2 + h
a >0 a 0 a

a =2,
y =− b 2a . y =0.
y =0.
y =0.
(0, 0).
(0, 0), (0, 0).
(2, 1) (8, 2) (2, −1), (8, −2).

x = y 2
x = −y 2
x =−y 2 +2y +8
a =−1,
y =− b 2a . y =1.
y =1.
y =1.

(9, 1).
y =0.
(8, 0).
(8, 0)
(8, 2)
(8, 2).
x =0.
x =0.
(0, 4) (0, −2).
x = −y 2 − 4y + 12
x = −y 2 + 2y − 3

x = a ⎛ ⎝y − k ⎞ ⎠2 + h.
x =2 ⎛ ⎝y −2 ⎞ ⎠2 +1
a =2, h =1, k =2
a =2,
y = k. y =2.
(h, k). (1, 2).
y =0. x = 2 ⎛ ⎝y −2 ⎞ ⎠2 +1
x = 2(0−2) 2 +1
x = 9
(9, 0).
(9, 0)
(9, 4)
x =0. x = 2 ⎛ ⎝y −2 ⎞ ⎠2 +1
0 = 2 ⎛ ⎝y −2 ⎞ ⎠2 +1
−1 = 2 ⎛ ⎝y −2 ⎞ 2
⎠

x = 3 ⎛
⎝y
− 1 ⎞2 ⎠ + 2
x = 2 ⎛
⎝y
− 3 ⎞2 ⎠ + 2
x =−4 ⎛ ⎝y +1 ⎞ ⎠2 +4
a =−4, h =4, k =−1
a =−4,
y = k.
(h, k).
y =−1.
(4, −1).
y =0. x = −4 ⎛ ⎝y +1 ⎞ ⎠2 +4
x = −4(0+1) 2 +4
x = 0

(0, 0).
(0, 0)
(0, −2)
x =−4 ⎛ ⎝y +1 ⎞ ⎠2 +4
x =0. 0 = −4 ⎛ ⎝y +1 ⎞ ⎠2 +4
−4 = −4 ⎛ ⎝y +1 ⎞ 2
⎠
1 = ⎛ ⎝y +1 ⎞ 2
⎠
y +1 = ±1
y =−1+1
y =−1−1
y =0 y =−2
(0, 0) (0, −2).
x = −4 ⎛
⎝y
+ 2 ⎞2 ⎠ + 4
x = −2 ⎛
⎝y
+ 3 ⎞2 ⎠ + 2
x =2y 2 +12y +17

x = a ⎛ ⎝y − k ⎞ ⎠2 + h
a =2, h =−1, k =−3
a =2,
y = k.
(h, k).
y =0.
y =−3.
(−1, −3).
x = 2 ⎛ ⎝y +3 ⎞ ⎠2 −1
x = 2(0+3) 2 −1
x = 17
(17, 0).
(17, 0)
x =0.
(17, −6)
x = 2 ⎛ ⎝y +3 ⎞ ⎠2 −1
0 = 2 ⎛ ⎝y +3 ⎞ ⎠2 −1
1 = 2 ⎛ ⎝y +3 ⎞ 2
⎠
1
2 = ⎛ ⎝y +3 ⎞ 2
⎠
y +3 = ± 1 2
y = −3± 2 2
y =−3+ 2 2
y =−3− 2 2
⎛ ⎝ 0, −3 + 2 2
y ≈ − 2.3 y ≈ − 3.7
⎞
⎠ , ⎛ ⎝ 0, −3 − 2
2 ⎞
⎠ .

a x = 3y 2 + 6y + 7 b
a
x = −4y 2 − 16y − 12 b
y = a(x − h) 2 + k.

(0, 0).
(20, 0)
(0, 0),
(20, 0).
(h, k). (h, k) = (10, 10)
h = 10, k =10
y = a(x − h) 2 + k
y = a(x −10) 2 +10
(x, y) = (0, 0)
y = a(x −10) 2 +10
0 = a(0−10) 2 +10
0 = a(0−10) 2 +10
−10 = a(−10) 2
−10 = 100a
−10
100 = a
a = − 1 10
y = a(x −10) 2 +10
y =− 1 10 (x −10)2 +10

y =−x 2 +4x −3 y =−x 2 +8x −15 y =6x 2 +2x −1
y =8x 2 −10x +3
a b
y =−x 2 +2x −4 y =2x 2 +4x +6 y =−2x 2 −4x −5
y =3x 2 −12x +7
x =−2y 2 x =3y 2 x =4y 2
x =−4y 2 x =−y 2 −2y +3 x =−y 2 −4y +5
x = y 2 +6y +8 x = y 2 −4y −12 x = ⎛ ⎝y −2 ⎞ ⎠2 +3
x = ⎛ ⎝y −1 ⎞ ⎠2 +4 x =− ⎛ ⎝y −1 ⎞ ⎠2 +2 x =− ⎛ ⎝y −4 ⎞ ⎠2 +3
x = ⎛ ⎝y +2 ⎞ ⎠2 +1 x = ⎛ ⎝y +1 ⎞ ⎠2 +2 x =− ⎛ ⎝y +3 ⎞ ⎠2 +2
x =− ⎛ ⎝y +4 ⎞ ⎠2 +3 x =−3 ⎛ ⎝y −2 ⎞ ⎠2 +3 x =−2 ⎛ ⎝y −1 ⎞ ⎠2 +2
x =4 ⎛ ⎝y +1 ⎞ ⎠2 −4 x =2 ⎛ ⎝y +4 ⎞ ⎠2 −2
a b
x = y 2 +4y −5 x = y 2 +2y −3 x =−2y 2 −12y −16
x =−3y 2 −6y −5

a b
c d e f

y = x 2
x = y 2

a
b

y = (x − 1) 2 − 2
x 2 − 8x = 8.
y = 2x 2 − 12x + 14

⎛
⎝(−c, 0), (c, 0) ⎞ ⎠
(x, y) d 1 + d 2
2a d 1 + d 2 =2a.
d 1 + d 2 = 2a
Use the distance formula to find d 1 , d 2 .
⎛
⎝x − (−c) ⎞ ⎠2 + ⎛ ⎝y −0 ⎞ ⎠2 + (x − c) 2 + ⎛ ⎝y −0 ⎞ ⎠2 = 2a
After eliminating radicals and simplifying,
we get:
To simplify the equation of the ellipse, we
let a 2 − c 2 = b 2 .
So, the equation of an ellipse centered at the
origin in standard form is:
x 2
a 2 + y2
a 2 − c 2 = 1
x 2
a 2 + y2
b 2 = 1
y-intercepts
x-intercepts
x 2
a 2 + y2
b 2 = 1
x 2
a 2 + y2
b 2 = 1
Let x =0. 0 2
a 2 + y2
b 2 = 1
Let y =0. x 2
a 2 + 02
b 2 = 1
y 2
b 2 = 1
x 2
a 2 = 1
y 2 = b 2
x 2 = a 2
y = ±b
x = ±a
The y-intercepts are (0, b) and (0, −b). The x-intercepts are (a, 0) and (−a, 0).
(0, 0)
(0, 0),

(a, 0) (−a, 0).
(0, b) (0, −b).
x 2
a 2 + y2
b 2 =1
0, 0
x 2
a 2 + y2
b 2 =1 a > b b > a
(−a, 0), (a, 0)
(0, −b), (0, b)
x 2
4 + y2
9 =1.

x 2
4 + y2
16 = 1.
x 2
9 + y2
16 = 1.
(0, 0).
x 2 +4y 2 = 16.
x 2 +4y 2 =16
x 2
16 + 4y2
16 = 16
16

x 2
16 + y2
4 =1
(0, 0).
16 > 4 x 2
a 2 = 16, a =±4
b 2 =4, b =±2
(4, 0), (−4, 0).
(0, 2), (0, −2).
9x 2 + 16y 2 = 144.
16x 2 + 25y 2 = 400.

a =4 a 2 =16
b =3 b 2 =9
x 2
a 2 + y2
b 2 =1
x 2
16 + y2
b 2 =1
x 2
16 + y2
9 =1
(h, k).
(x − h)2
a 2
+ ⎛ ⎝y − k ⎞ 2
⎠
b 2 =1 a > b,
b > a,

(h, k)
(h, k),
(x − h) 2
a 2
+ ⎛ ⎝y − k ⎞ 2
⎠
b 2 =1
a > b,
b > a,
(x −3)2
9
+ ⎛ ⎝y −1 ⎞ 2
⎠
=1.
4
(x − h) 2
a 2 + ⎛ ⎝y − k ⎞ 2
⎠
b 2 =1.
(x −3) 2
9
+ ⎛ ⎝y −1 ⎞ 2
⎠
=1
4
(h, k). (3, 1).
9>4 x 2
a 2 =9, a =±3
b 2 =4, b =±2
(x + 3)2
4
+ ⎛
⎝y
− 5 ⎞2
⎠
= 1.
16
(x − 1)2
25
+ ⎛
⎝y
+ 3 ⎞2
⎠
= 1.
16
x2
9 + y2 (x −3)2
=1
4 9
+ ⎛ ⎝y −1⎞ 2
⎠
=1, a =3
4
b =2.

(x −3)2
9
+ ⎛ ⎝y −1 ⎞ 2
⎠
=1
4
(x +4)2
16
+ ⎛ ⎝y −6 ⎞ 2
⎠
=1
9
16 x2 + y2 =1 (0, 0).
9
(0, 0). (0, 0)
16 > 9,
a 2 = 16, a =±4
b 2 =9, b =±3
(4, 0), (−4, 0).
(0, 3), (0, −3).

(x − h) 2
a 2 + ⎛ ⎝y − k ⎞ 2
⎠
b 2 =1.
(x − (−4)) 2
16
+ ⎛ ⎝y −6 ⎞ 2
⎠
=1
9
(h, k). (−4, 6).
16 x2 + y2 =1
9
(−4, 6).
(x +4) 2
16
+ ⎛ ⎝y −6⎞ 2
⎠
=1.
9
(x − 5)2
9
+ ⎛
⎝y
+ 4 ⎞2
⎠
= 1
4
(x + 6)2
16
+ ⎛
⎝y
+ 2 ⎞2
⎠
= 1
25
x 2 y 2
x 2 +4y 2 −4x +24y +24=0

x 2 +4y 2 −4x +24y +24=0
x 2 y 2
(x − h) 2
a 2 + ⎛ ⎝y − k ⎞ 2
⎠
b 2 =1
(h, k).
16 > 4 x 2
a 2 = 16, a =±4
b 2 =4, b =±2
(2, −3).
a 6x 2 + 4y 2 + 12x − 32y + 34 = 0 b
a 4x 2 + y 2 − 16x − 6y + 9 = 0 b

a =40 a 2 = 1600
b
(10, 0) c =10 b 2
x 2
a 2 + y2
b 2 =1
x 2
1600 + y2
b 2 =1
b 2 = a 2 − c 2
b 2 =40 2 −10 2
b 2 = 1600 − 100
b 2 = 1500
a 2 b 2 x 2
1600 + y2
1500 =1

x 2
4 + y2
25 =1 x 2
9 + y2
25 =1 x 2
25 + y2
36 =1
x 2
16 + y2
36 =1 x 2
36 + y2
16 =1 x 2
25 + y2
9 =1
x 2 + y2
4 =1 x 2
9 + y2 =1 4x 2 +25y 2 = 100
16x 2 +9y 2 = 144 16x 2 +36y 2 = 576 9x 2 +25y 2 = 225

(x +1) 2
4
+ ⎛ ⎝y +6 ⎞ 2
⎠
=1
25
(x −3) 2
25
+ ⎛ ⎝y +2 ⎞ 2
⎠
=1
9
(x +4) 2
4
+ ⎛ ⎝y −2 ⎞ 2
⎠
=1
9
(x −4) 2
9
+ ⎛ ⎝y −1 ⎞ 2
⎠
=1
16
(x −3) 2
4
+ ⎛ ⎝y −7 ⎞ 2
⎠
=1
25
(x +6) 2
16
+ ⎛ ⎝y +5 ⎞ 2
⎠
=1
4
(x −5) 2
9
+ ⎛ ⎝y +4 ⎞ 2
⎠
=1
25
(x +5) 2
36
+ ⎛ ⎝y −3 ⎞ 2
⎠
=1
16
a b
25x 2 +9y 2 − 100x −54y −44=0 4x 2 +25y 2 +8x + 100y +4=0
4x 2 +25y 2 −24x −64=0 9x 2 +4y 2 +56y + 160 = 0
x =−2 ⎛ ⎝y −1 ⎞ ⎠2 +2 x 2 + y 2 =49 (x +5) 2 + ⎛ ⎝y +2 ⎞ ⎠2 =4
y =−x 2 +8x −15
(x +3) 2
16
+ ⎛ ⎝y +1 ⎞ 2
⎠
=1
4
(x −2) 2 + ⎛ ⎝y −3 ⎞ ⎠2 =9
x 2
25 + y2
36 =1 x =4 ⎛ ⎝y +1 ⎞2 ⎠ −4 x 2 + y 2 =64
x 2
9 + y2
25 =1 y =6x 2 +2x −1
(x −2) 2
9
+ ⎛ ⎝y +3 ⎞ 2
⎠
=1
25

x 2
4 + y2 =1
x2
9 9 + y2
4 =1.

a
b

(0, 0)
(h, k)
x 2 = 12.
(x − 4) 2 .
y = − 2 3 x.

⎛ ⎝(−c, 0), (c, 0) ⎞ ⎠
(x, y) | d 1 − d 2|
2a | d 1 − d 2| =2a.
|d 1 − d 2| =2a
Use the distance formula to find d 1 , d 2 |
(x − (−c)) 2 + ⎛ ⎝y −0 ⎞ ⎠2 − (x − c) 2 + ⎛ ⎝y −0 ⎞ ⎠2
|
=2a
Eliminate the radicals.
To simplify the equation of the ellipse, we
let c 2 − a 2 = b 2 .
So, the equation of a hyperbola centered at
the origin in standard form is:
x 2
a 2 + y2
c 2 − a 2 =1
x 2
a 2 − y2
b 2 =1
x-intercepts
y-intercepts
x 2
a 2 − y2
b 2 = 1 x 2
a 2 − y2
b 2 = 1
Let y =0. x 2
a 2 − 02
b 2 = 1 Let x =0. 0 2
a 2 − y2
b 2 = 1
x 2
a 2 = 1 − y2
b 2 = 1
x 2 = a 2 y 2 = −b 2
x = ±a y = ± −b 2
The x-intercepts are (a, 0) and (−a, 0).
There are no y-intercepts.

(−a, 0), (a, 0)
(0, −b), (0, b).
y = b a x y =−b a x.
(0, 0)
(0, 0),
x 2
a 2 − y2
b 2 =1 or y 2
a 2 − x2
b 2 =1
x 2 a 2 y 2
b 2 .
x 2 y 2

0, 0
x 2
a 2 − y2
b 2 =1 y 2
a 2 − x2
b 2 =1
(−a, 0), (a, 0) (0, −a), (0, a)
(−a, 0), (a, 0)
(0, −a), (0, a)
(±a, 0) (0, ±b) (0, ±a) (±b, 0)
y = b a x, y =−b a x y = a b x, y =−a b x
(0, 0)
x2
25 − y2
4 =1.

x2
16 − y2
4 = 1.
x2
9 − y2
16 = 1.
(0, 0).
±a ±b.
4y 2 −16x 2 = 64.

4y 2 −16x 2 =64
4y 2
64 − 16x2
64 = 64
64
y 2
16 − x2
4 =1
a 2 =16 a =±4.
(0, −a),
b 2 =4 b =±2.
(0, a).
(0, −4), (0, 4)
(−2, 0), (2, 0)
4y 2 − 25x 2 = 100.
25y 2 − 9x 2 = 225.
(h, k)
(h, k)

h, k
x − h 2
a 2
− y − k 2
b 2 =1
y − k 2
a 2
x − h2
−
b 2 =1
(h, k) (h, k)
(h, k)
(x −1)2
9
− ⎛ ⎝y −2 ⎞ 2
⎠
=1
16

(x − 3)2
25
− ⎛
⎝y
− 1 ⎞2
⎠
= 1.
9
(x − 2)2
4
− ⎛
⎝y
− 2 ⎞2
⎠
= 1.
9
(h, k).
(h, k)
x − h y − k (h, k).
⎛
⎝y +2 ⎞ 2
⎠
−
9
(x +1)2
4
=1.

y 2 -
(h, k). (−1, −2)
a =3 b =2
⎛
⎝y
+ 3 ⎞ ⎠ ⎞2
16
−
(x + 2)2
9
= 1.
⎛
⎝y
+ 2 ⎞ ⎠ ⎞2
9
−
(x + 2)2
9
= 1.
4x 2 −9y 2 −24x −36y −36=0.

x 2 -
(h, k). (3, −2)
a =3
b =4
a b 9x 2 − 16y 2 + 18x + 64y − 199 = 0.
a b 16x 2 − 25y 2 + 96x − 50y − 281 = 0.

x 2 - y 2 -
x 2 y 2 . x =3y 2 −2y +1
x 2 - y 2 - x 2 + y 2 =49
x 2 - y 2 - 4x 2 +25y 2 = 100
x 2 - y 2 - 25y 2 −4x 2 = 100
a 9x 2 +4y 2 +56y + 160 = 0 b 9x 2 −16y 2 +18x +64y − 199 = 0 c x 2 + y 2 −6x −8y =0 d
y =−2x 2 −4x −5
a
9x 2 +4y 2 +56y + 160 = 0
x 2 y 2
b
9x 2 −16y 2 +18x +64y − 199 = 0
x 2 y 2
c
x 2 + y 2 −6x −8y =0
x 2 y 2
d
y =−2x 2 −4x −5
x

a x 2 + y 2 − 8x − 6y = 0 b 4x 2 + 25y 2 = 100 c y = 6x 2 + 2x − 1 d 16y 2 − 9x 2 = 144
a 16x 2 + 9y 2 = 144 b y = 2x 2 + 4x + 6 c x 2 + y 2 + 2x + 6y + 9 = 0 d 4x 2 − 16y 2 = 64

(0, 0)
x 2
9 − y2
4 =1 x 2
25 − y2
9 =1 x 2
16 − y2
25 =1
x 2
9 − y2
36 =1 y 2
25 − x2
4 =1 y 2
36 − x2
16 =1
16y 2 −9x 2 = 144 25y 2 −9x 2 = 225 4y 2 −9x 2 =36
16y 2 −25x 2 = 400 4x 2 −16y 2 =64 9x 2 −4y 2 =36
(h, k)
(x −1) 2
16
− ⎛ ⎝y −3 ⎞ 2
⎠
=1
4
(x −2) 2
4
− ⎛ ⎝y −3 ⎞ 2
⎠
=1
16
⎛
⎝y −4 ⎞ 2
⎠
−
9
(x −2)2
25
=1
⎛
⎝y −1 ⎞ 2
⎠
−
25
(x −4)2
16
=1
⎛
⎝y +4 ⎞ 2
⎠
−
25
(x +1)2
36
=1
⎛
⎝y +1 ⎞ 2
⎠
−
16
(x +1)2
4
=1
⎛
⎝y −4 ⎞ 2
⎠
−
16
(x +1)2
25
=1
⎛
⎝y +3 ⎞ 2
⎠
−
16
(x −3)2
36
=1
(x −3) 2
25
− ⎛ ⎝y +2 ⎞ 2
⎠
=1
9
(x +2) 2
4
− ⎛ ⎝y −1 ⎞ 2
⎠
=1
9
a b
9x 2 −4y 2 −18x +8y −31=0 16x 2 −4y 2 +64x −24y −36=0
y 2 − x 2 −4y +2x −6=0 4y 2 −16x 2 −24y +96x − 172 = 0
9y 2 − x 2 +18y −4x −4=0
a x =−y 2 −2y +3 b a x =−2y 2 −12y −16 b x 2 + y 2 =9 c
9y 2 − x 2 +18y −4x −4=0 c 9x 2 +25y 2 = 225 16x 2 −4y 2 +64x −24y −36=0
d x 2 + y 2 −4x +10y −7=0
16x 2 +36y 2 = 576
d

⎛
⎝y −3 ⎞ 2
⎠ (x +2)2
x
− =1
2 + y 2 −4x +10y −7=0 y = (x −1) 2 +2
9 16
x 2
9 + y2
25 =1 (x +2) 2 + ⎛ ⎝y −5 ⎞2 ⎠ =4 y 2 − x 2 −4y +2x −6=0
x =−y 2 −2y +3 16x 2 +9y 2 = 144
x 2
4 − y2 y2
=1
9 9 − x2
4 =1.
a
b

⎧x
− 3y = −3
⎨ .
⎩ x + y = 5
⎧x
− 4y = −4
⎨
⎩−3x + 4y = 0 .
⎧3x − 4y = −9
⎨
⎩5x + 3y = 14 .
⎧
⎨ x2 + y 2 =9
⎩x 2 − y =9
⎧
⎨ 9x2 + y 2 =9
⎩y =3x −3
⎧ x + y =4
⎨
⎩y = x 2 +2

⎧ x − y =−2
⎨
⎩y = x 2 .
⎧
⎨ x − y = −2
⎩y = x 2
line
parabola
x − y =−2.
y = x +2.
y = x 2 .
(2, 4) (−1, 1).

(2, 4)
x − y = −2 y = x 2
2−4 = ? −2 4 = ? 2 2
−2 = −2 ✓ 4 = 4 ✓
(−1, 1)
x − y = −2 y = x 2
−1−1 = ? −2 1 = ? (−1) 2
−2 = −2 ✓ 1 = 1 ✓
(2, 4) (−1, 1).
⎧ x + y = 4
⎨
⎩ y = x 2 + 2 .
⎧x
− y = −1
⎨
⎩ y = −x 2 + 3 .
x 2 y 2
⎧ y =−1 ⎨
⎩(x −2) 2 + ⎛ ⎝y +3 ⎞2 ⎠ =4 .
⎧
⎨ y = −1
line
⎩(x −2) 2 + ⎛ ⎝y +3 ⎞ ⎠2 = 4 circle

(x −2) 2 + ⎛ ⎝y +3 ⎞ ⎠2 =4
(2, −3)
y =−1.
(2, −1).
(2, −1)
(x −2) 2 + ⎛ ⎝y +3 ⎞ ⎠2 = 4 y = −1
(2−2) 2 + (−1 + 3) 2 = ? 4 −1 = −1 ✓
(0) 2 + (2) 2 = ? 4
4 = 4 ✓
(2, −1).
x = −6
(x + 3) 2 + ⎛
⎝y
− 1 ⎞2 ⎠ = 9 .
⎧
⎨
⎩
⎧ y = 4
⎨
⎩(x − 2) 2 + ⎛
⎝y
+ 3 ⎞2 ⎠ = 4 .

⎧
⎨ 9x2 + y 2 =9
⎩y =3x −3 .
⎧
⎨ 9x2 + y 2 = 9
⎩y =3x − 3
ellipse
line
y =3x −3
3x −3
x =0 x =1 y =3x −3
(0, −3), (1, 0).

(0, −3)
9x 2 + y 2 = 9 y = 3x −3
9·0 2 + (−3) 2 = ? 9 −3 = ? 3·0−3
(1, 0)
0+9 = ? 9 −3 = ? 0−3
9 = 9 ✓ −3 = −3 ✓
9x 2 + y 2 = 9 y = 3x −3
9·1 2 +0 2 = ? 9 0 = ? 3·1−3
9+0 = ? 9 0 = ? 3−3
9 = 9 ✓ 0 = 0 ✓
(0, −3), (1, 0).
⎧x
2 + 9y 2 = 9
⎨
⎩ y = 1 3 x − 3 .
⎧4x 2 + y 2 = 4
⎨
.
⎩ y = x + 2
⎧
⎨ x2 − y =0
⎩y = x −2 .
⎧
⎨ x2 − y = 0
⎩y = x − 2
parabola
line
y = x −2
x −2

2 −4ac
(−1) 2 −4·1·2
−7
⎧x
2 − y = 0
⎨
⎩ y = 2x − 3 .
⎧y
2 − x = 0
⎨
⎩ y = 3x − 2 .
⎧
⎨ x2 + y 2 =4
⎩x 2 − y =4 .

x 2 ,
−1.
x 2 .
y =0 y =−1
(−2, 0) (2, 0).
⎛
⎝
3, −1 ⎞ ⎠ ⎛ ⎝−3,−1 ⎞ ⎠
(−2, 0), (2, 0),
⎛
⎝− 3,−1 ⎞ ⎠.
⎛
⎝ 3, −1 ⎞ ⎠,
⎧x
2 + y 2 = 9
⎨
⎩x
2 − y = 9 .
⎧x
2 + y 2 = 1
⎨
⎩−x + y 2 = 1 .

⎧
⎨ x2 + y 2 =7
⎩x 2 − y 2 =1 .
⎧
⎨ x2 + y 2 = 7
⎩x 2 − y 2 = 1
circle
hyperbola
y 2
⎧
⎨ x2 + y 2 =7
⎩x 2 − y 2 =1
⎧
⎨ x2 + y 2 =7
⎩__________
x 2 − y 2 =1
2x 2 =8
x 2 = 4
x = ±2
x =2 x =−2
x =2 x =−2
x 2 + y 2 = 7 x 2 + y 2 = 7
2 2 + y 2 = 7 (−2) 2 + y 2 = 7
4+y 2 = 7 4+y 2 = 7
y 2 = 3 y 2 = 3
y = ± 3 y = ± 3
⎛ ⎝−2, 3 ⎞ ⎠,
⎛
⎝2, 3 ⎞ ⎠, ⎛ ⎝2, − 3 ⎞ ⎠.
⎛
⎝−2, − 3 ⎞ ⎠,

⎛ ⎝−2, 3 ⎞ ⎠,
⎛ ⎝2, − 3 ⎞ ⎠.
⎛
⎝−2, − 3 ⎞ ⎠,
⎛
⎝2, 3 ⎞ ⎠,
⎧x
2 + y 2 = 25
⎨
⎩y
2 − x 2 = 7 .
⎧x
2 + y 2 = 4
⎨
⎩x
2 − y 2 = 4 .
x =
y =

−20.
−1.
x =
y =

y 2

⎧y =2x +2
⎧y =6x −4
⎨
⎩y =−x 2 ⎨
+2
⎩y =2x 2
⎧ x + y =2
⎨
⎩x = y 2
⎧ x − y =−2
⎨
⎩x = y 2
⎧
⎨ y = 3 2 x +3
⎩y =−x 2 +2
⎧ y = x −1
⎨
⎩y = x 2 +1
⎧ x =−2
⎨
⎩x 2 + y 2 =4
⎧ y =−4 ⎨
⎩x 2 + y 2 =16
⎧ x =2
⎨
⎩(x +2) 2 + ⎛ ⎝y +3 ⎞ ⎠2 =16
⎧ y =−1 ⎨
⎩(x −2) 2 + ⎛ ⎝y −4 ⎞ ⎠2 =25
⎧
⎨
⎩
y =−2x +4
y = x +1
⎧
⎨ y =−1 2 x +2
⎩y = x −2
⎧
⎨ x2 +4y 2 =4
y = 1 2 x −1
⎩
⎧
⎨ 9x2 + y 2 =9
⎩y =3x +3
⎧
⎨ 9x2 + y 2 =9
⎩y = x +3
⎧
⎨ 9x2 +4y 2 =36
⎩x =2
⎧
⎨ 4x2 + y 2 =4
⎩y =4
⎧
⎨ x2 + y 2 = 169
⎩x =12
⎧
⎨ 3x2 − y =0
⎩y =2x −1
⎧
⎨ 2y2 − x =0
⎩y = x +1
⎧
⎨ y = x2 +3
⎩y = x +3
⎧
⎨ y = x2 −4
⎩y = x −4
⎧
⎨ x2 + y 2 =25
⎩x − y =1
⎧
⎨ x2 + y 2 =25
⎩2x + y =10
⎧
⎨ x2 + y 2 =16
⎧
⎩x 2 ⎨ x2 + y 2 =16
−2y =8
⎩x 2 − y =4
⎧
⎨ x2 + y 2 =4
⎩x 2 +2y =1
⎧
⎨ x2 + y 2 =4
⎩x 2 − y =2
⎧
⎨ x2 + y 2 =9
⎩x 2 − y =3
⎧
⎨ x2 + y 2 =4
⎩y 2 − x =2
⎧
⎨ x2 + y 2 =25
⎩2x 2 −3y 2 =5
⎧
⎨ x2 + y 2 =20
⎩x 2 − y 2 = −12
⎧
⎨ x2 + y 2 =13
⎩x 2 − y 2 =5

⎧
⎨ x2 + y 2 =16
⎩x 2 − y 2 =16
⎧
⎨ 4x2 +9y 2 =36
⎩2x 2 −9y 2 =18
⎧
⎨ x2 − y 2 =3
⎩2x 2 + y 2 =6
⎧
⎨ 4x2 − y 2 =4
⎩4x 2 + y 2 =4
⎧
⎨ x2 − y 2 =−5
⎩3x 2 +2y 2 =30
⎧
⎨ x2 − y 2 =1
⎩x 2 −2y =4
⎧
⎨ 2x2 + y 2 =11
⎩x 2 +3y 2 =28
−6
−42.
cm 2 .

a
b

(x 1 , y 1 ) (x 2 , y 2 )
d = (x 2 − x 1 ) 2 + (y 2 − y 1 ) 2
(x 1 , y 1 ) (x 2 , y 2 )
⎛
⎝
x 1 + x 2
2
, y 1 + y 2⎞
2 ⎠
(h, k),
(h, k),
x 2 + y 2 + ax + by + c =0

y = ax 2 + bx + c
y = ax − h 2 + k
a >0 a 0 a

x = ay 2 + by + c
x = a y − k 2 + h
a >0 a 0 a

(0, 0) :
(0, 0),
x 2
a 2 + y2
b 2 =1
(a, 0) (−a, 0).
(0, b) (0, −b).
(0, 0)
(h, k) :
(h, k),
(x − h) 2
a 2
+ ⎛ ⎝y − k ⎞ 2
⎠
b 2 =1
a > b,
b > a,

0, 0
x 2
a 2 − y2
b 2 =1 y 2
a 2 − x2
b 2 =1
(−a, 0), (a, 0) (0, −a), (0, a)
(−a, 0), (a, 0)
(0, −a) (0, a)
(±a, 0) (0, ±b) (0, ±a) (±b, 0)
y = b a x, y =−b a x y = a b x, y =−a b x
(0, 0).
±a ±b.
h, k
x − h 2
a 2
− y − k 2
b 2 =1
y − k 2
a 2
x − h2
−
b 2 =1
(h, k) (h, k)
(h, k).
a, b.
(h, k) a, b.

x 2 - y 2 -
x 2 y 2 . x =3y 2 −2y +1
x 2 - y 2 - x 2 + y 2 =49
x 2 - y 2 - 4x 2 +25y 2 = 100
x 2 - y 2 - 25y 2 −4x 2 = 100
(−5, 1) (−1, 4) (−2, 5) (1, 5) (8, 2) (−7, −3)

(1, −4) (5, −5)
(−2, −6) (−4, −2) (3, 7) (5, 1) (−8, −10) (9, 5)
(−3, 2) (6, −9)
7
(0, 0)
(0, 0)
(−3, 5)
(−2, −5)
(3, 6)
(3, −2)
(2, 2)
(4, 4)
a b
2x 2 +2y 2 = 450 3x 2 +3y 2 = 432 (x +3) 2 + ⎛ ⎝y −5 ⎞ ⎠2 =81
(x +2) 2 + ⎛ ⎝y +5 ⎞2 ⎠ =49
x 2 + y 2 −6x −12y −19=0
x 2 + y 2 −4y −60=0
y = x 2 +4x −3 y =2x 2 +10x +7
y =−6x 2 +12x −1 y =−x 2 +10x
a b
y = x 2 +4x +7 y =2x 2 −4x −2
y =−3x 2 −18x −29 y =−x 2 +12x −35
x =2y 2 x =2y 2 +4y +6
x =−y 2 +2y −4 x =−3y 2

a b
x =4y 2 +8y x = y 2 +4y +5
x =−y 2 −6y −7 x =−2y 2 +4y
x 2
36 + y2
25 =1 x 2
4 + y2
81 =1
49x 2 +64y 2 = 3136 9x 2 + y 2 =9
(x −1) 2
25
+ ⎛ ⎝y −6 ⎞ 2
⎠
=1
4
(x +4) 2
16
+ ⎛ ⎝y +1 ⎞ 2
⎠
=1
9
(x −5) 2
16
+ ⎛ ⎝y +3 ⎞ 2
⎠
=1
36
(x +3) 2
9
+ ⎛ ⎝y −2 ⎞ 2
⎠
=1
25
a b
4x 2 +16y 2 +48x + 160y + 480 = 0 25x 2 +4y 2 − 150x −56y + 321 = 0

25x 2 +4y 2 + 150x + 125 = 0 4x 2 +9y 2 − 126y + 405 = 0
(0, 0)
x 2
25 − y2
9 =1 y 2
49 − x2
16 =1
9y 2 −16x 2 = 144 16x 2 −4y 2 =64
(h, k)
(x +1) 2
4
− ⎛ ⎝y +1 ⎞ 2
⎠
=1
9
(x −2) 2
4
− ⎛ ⎝y −3 ⎞ 2
⎠
=1
16
⎛
⎝y +2 ⎞ 2
⎠
−
9
(x +1)2
9
=1
⎛
⎝y −1 ⎞ 2
⎠
−
25
(x −2)2
9
=1
a b
4x 2 −16y 2 +8x +96y − 204 = 0 16x 2 −4y 2 −64x −24y −36=0
4y 2 −16x 2 +32x −8y −76=0 36y 2 −16x 2 −96x + 216y − 396 = 0

a 16y 2 −9x 2 −36x −96y −36=0
b x 2 + y 2 −4x +10y −7=0
c y = x 2 −2x +3
d 25x 2 +9y 2 = 225
a x 2 + y 2 +4x −10y +25=0
b y 2 − x 2 −4y +2x −6=0
c x =−y 2 −2y +3
d 16x 2 +9y 2 = 144
⎧
⎨ 3x2 − y =0
⎧
⎨ y = x2 −4
⎩y =2x −1
⎩y = x −4
⎧
⎨ x2 + y 2 = 169
⎩x =12
⎧
⎨ x2 + y 2 =25
⎩y =−5
⎧
⎨ y = x2 +3
⎧
⎨ x2 + y 2 =4
⎩y =−2x +2
⎩x − y =4
⎧
⎨ 9x2 +4y 2 =36
⎩y − x =5
⎧
⎨ x2 +4y 2 =4
⎩2x − y =1
⎧
⎨ x2 + y 2 =16
⎧
⎩x 2 ⎨ x2 − y 2 =5
−2y −1=0
⎩−2x 2 −3y 2 = −30
⎧
⎨ 4x2 +9y 2 =36
⎩3y 2 −4x =12
⎧
⎨ x2 + y 2 =14
⎩x 2 − y 2 =16

(−4, −3) (−10, −11) (6, 8) (−5, −3)
(0, 0) (10, −2)
(2, −3)
(−2, 3)
a b
4x 2 +49y 2 = 196 y =3(x −2) 2 −2 3x 2 +3y 2 =27
y 2
100 − x2
36 =1 x 2
16 + y2
81 =1 x =2y 2 +10y +7
64x 2 −9y 2 = 576
a b
c
25x 2 +64y 2 + 200x − 256y − 944 = 0 x 2 + y 2 +10x +6y +30=0
x =−y 2 +2y −4 9x 2 −25y 2 −36x −50y − 214 = 0
y = x 2 +6x +8
⎧
⎨ 3y2 − x =0
⎩y =−2x −1 .
⎧
⎨ x2 + y 2 =8
⎩y =−x −4 . ⎧
⎨ x2 +9y 2 =9
⎩2x 2 −9y 2 =18 .

y = ax 2 + bx + c
−240.

2n + 3
(−1) n

f (n) = n 2 + 2, f (1) + f (2) + f (3).
f (x) =2x
f(x) =2x
x
2x
2, 4, 6, 8, 10, …,
2, 4, 6, 8
f (x) =2x 2x
f (x) =2x, a n =2n. a n
a n =2n,

n a n
a n 2·1
2
2·2
4
2·3
6
2·4
8
2·5
10
a 1 , a 2 , a 3 , a 4 , a 5 ,…, a n ,…
2, 4, 6, 8, 10, …
a n =4n −3.
a n =4n −3,
a n = 3n − 4.
a n = 2n − 5.
a n =2 n +1.
a n =2 n +1,
a n = 3 n + 4.
a n = 2 n − 5.
(−1) n (−1) n +1

n
(−1) n (−1) 1
(−1) 2
(−1) 3
(−1) 4
(−1) 5
−1
1
−1
1
−1
(−1) n +1 (−1) 1+1
(−1) 2+1
(−1) 3+1
(−1) 4+1
(−1) 5+1
1
−1
1
−1
1
a 1 , a 2 , a 3 , a 4 , a 5 ,…, a n ,…
−1, 1, −1, 1, −1…
1, −1, 1, −1, 1…
−1.
a n = (−1) n n 3 .
a n = (−1) n n 3 ,
−1, 8, −27, 64, −125.
a n = (−1) n n 2 .
a n = (−1) n + 1 n 3 .
4, 8, 12, 16, 20, …
a n =4n.

3, 6, 9, 12, 15, …
5, 10, 15, 20, 25, …
2, −4, 8, −16, 32, …
n
a n = (−1) n +1 2 n .
−3, 9, −27, 81, −243, …
1, −4, 9, −16, 25, …
1
3 , 1 9 , 1 , 1 , 1
27 81 243 ,…
a n = 1 3 n.

1
2 , 1 4 , 1 8 , 1
16
, 1
32
, …
1
1 , 1 4 , 1 9 , 1
16
, 1
25
, …
5! 5·4·3·2·1.
n!
0! 0!=1.
n! =n(n −1)(n −2) …
n!
1! 2! 3! 4! 5!
1 2⋅1 3⋅2⋅1 4⋅3⋅2⋅1 5⋅4⋅3⋅2⋅1
1 2 6 24 120
a n = 1 n!
a n = 1 n! ,
1, 1 2 , 1 6 , 1
24
, 1
120 .
a n = 2 n! .
a n = 3 n! .

(n +1)!
a n =
(n −1)! .
a n =
(n +1)!
(n −1)! ,
a n =
(n − 1)!
(n + 1)! .
a n = n!
(n + 1)! .
5
a 1 + a 2 + a 3 + a 4 + a 5 ∑ a i .
i =1
∑
a n
n
∑ a i = a 1 + a 2 + a 3 + a 4 + a 5 +…+a n
i =1
5
∑ 2i.
i =1

5
∑ 2i
i =1
2·1+2·2+2·3+2·4+2·5
2+4+6+8+10
30
5
∑ 2i =30
i =1
5
∑ 3i.
i = 1
5
∑ 4i.
i = 1
3
∑ 1
k ! .
k =0
3
∑ 1
k !
k =0
1
0! + 1 1! + 1 2! + 1 3!
1
1 + 1 1 + 1 2 + 1 6
1+1+ 3 6 + 1 6
16
6
8
3
3
∑ 1
k ! = 8 3
k =0

3
∑ 2
k ! .
k = 0
3
∑ 3
k ! .
k = 0
1+ 1 2 + 1 3 + 1 4 + 1 5 .
1+ 1 2 + 1 3 + 1 4 + 1 5
n: 1,2,3,4,5
1, 1 2 , 1 3 , 1 4 , 1 5
1 1 , 1 2 , 1 3 , 1 4 , 1 5 , … 1 n
1+ 1 2 + 1 3 + 1 4 + 1 5 = ∑
5
1
n .
n =1
1
2
+ 1 4 + 1 8 + 1
16 + 1 32 .
1 + 1 4 + 1 9 + 1 16 + 1 25 .
−1 + 8 − 27 + 64 − 125.

5
−1 + 8 − 27 + 64 − 125 = ∑ (−1) n ⋅ n 3
n =1
1 − 4 + 9 − 16 + 25.
−2 + 4 − 6 + 8 − 10.

a n =2n −7
a n =5n −1
a n =3n +1 a n =4n +2
a n =2 n +3
a n =3 n −2n
a n =3 n −1
a n =2 n −3n
a n = 2n
n 2
a n = 3n
n 3
a n =
4n −2
2 n a n = 3n +3
3 n
a n = (−1) n ·2n a n = (−1) n ·3n
a n = (−1) n +1 n 2 a n = (−1) n +1 n 4
+1
(−1)n
a n =
n 2 a n =
(−1)n
+1
2n
8, 16, 24, 32, 40, …
7, 14, 21, 28, 35, …
6, 7, 8, 9, 10, …
−3,−2,−1,0,1,…
e 3 , e 4 , e 5 , e 6 , e 7 ,… 1
e 2, 1 e ,1,e, e2 ,…
−5, 10, −15, 20, −25, …
−1, 8, −27, 64, −125, …
−2, 4, −6, 8, −10, …
−6, 11, −16, 21, −26, …
2, −5, 10, −17, 26, …
1, −3, 5, −7, 9, …
1
4
, 1
16
, 1
64
, 1
256 , 1
1,024 ,… 1
1
, 1 8 , 1
27
, 1
64
, 1
125 ,…
− 1 2 , − 2 3 , − 3 4 , − 4 5 , − 5 6 ,…
−2, − 3 2 , − 4 3 , − 5 4 , − 6 5 ,…
− 5 2 , − 5 4 , − 5 8 , − 5 16 , − 5 32 ,… 4, 1 2 , 4
27
, 4
64
, 4
125 ,…

a n = 4 n!
a n = 5 n!
a n =3n!
a n = (2n)!
a n =
a n = n!
n 2
(n −1)!
(n)!
a n =2n!
a n = (3n)!
a n = n!
(n +1)!
a n = n2
n!
a n =
(n +1)!
(n +1)!
n 2 a n =
2n
5
∑ i 2
5
∑ i 3
i =1
i =1
6
6
∑ (2i +3) ∑ (3i −2)
i =1
i =1
4
∑ 2 i 4
∑ 3 i
i =1
i =1
3
∑ 4
k !
k =0
4
∑ −
k 1 !
k =0
5
5
∑ k(k +1) ∑ k(2k −3)
k =1
k =1
5
∑ n
n +1
n =1
4
∑ n
n +2
n =1
1
3
+ 1 9 + 1 27 + 1 81 + 1
243
1
4
+ 1 16 + 1 64 + 1
256
1+ 1 8 + 1 27 + 1 64 + 1
125
1
5
+ 1 25 + 1
125 + 1
625
2+1+ 2 3 + 1 2 + 2 5
3+ 3 2 +1+3 4 + 3 5 + 1 2

3−6+9−12+15
−5+10−15+20−25
−2+4−6+8−10+ … +20 1−3+5−7+9+ … +21
14 + 16 + 18 + 20 + 22 + 24 + 26 9 + 11 + 13 + 15 + 17 + 19 + 21
a n = (−1) n (n +2)?
n!
12
∑ 2k
k =1
a
b

n
n
4n − 1
⎧ x + y = 7
⎨
⎩3x + 4y = 23 .
f (n) = n (3n + 5), f (1) + f (20).
2
a n − a n −1 ,
a n − a n −1 ,
a 5, 9, 13, 17, 21, 25, …
b 4, 9, 12, 17, 20, 25, …

c 10, 3, −4, −11, −18, −25, …
a
5, 9, 13, 17 21, 25, …
9−5 13−9 17−13 21−17 25−21
4 4 4 4 4
d =4.
b
4, 9, 12, 17 20, 25, …
9−4 12−9 17−12 20−17 25−20
2 3 5 3 5
c
10, 3, −4, −11 −18, −25, …
3 − 10 −4 − 3 −11 − (−4) −18 − (−11) −25 − (−18)
−7 −7 −7 −7 −7
d =−7.
a 9, 20, 31, 42, 53, 64, … b 12, 6, 0, −6, −12, −18, … c 7, 1,10, 4, 13, 7, …
a −4, 4, 2, 10, 8, 16, … b −3, −1, 1, 3, 5, 7, … c 7, 2, −3, −8, −13, −18, …
a 1 ,
d =−6.

a 1 a 2 a 3 a 4 a 5
5 5+(−6) −1 + (−6) −7 + (−6) −13 + (−6)
−1 −7 −13 −19
5, −1, −7, −13, −19, …
d = −4.
d = −8.
a 1
a 1
a 1
a 1
a n = a 1 + (n −1)d
a 1
a n = a 1 + (n −1)d

a 15 ,
a 1 = 3 and d =6.
a n = a 1 + (n −1)d
a 15 =3+(15 − 1)6
a 15 =3+(14)6
a 15 =87
−7
−2

a 1 ,
a 7 = 10, n = 7, and d =−2.
a n = a 1 + (n −1)d
10 = a 1 + (7−1)(−2)
10 = a 1 + (6)(−2)
10 = a 1 −12
a 1 =22
a 12 ,
a 1 = 22, n = 12, and d =−2.
a n = a 1 + (n −1)d
a 12 =22+(12 − 1)(−2)
a 12 =22+(11)(−2)
a 12 =0
a 12 =0.
a n = a 1 + (n −1)d
a n =22+(n −1)(−2)
a n =22−2n +2
a n =−2n + 24.
−3.
−4.

a 5 a 11 ,
n = 11.
n =5
a 5 =19 a 11 = 37.
a 1
−1.
d =3
a 1 .
a 1 =7 d =3.
a 1 =7.
d =3.
a n =3n +4.
−25.
S n ,
n
S n = a 1 + a 2 + a 3 + ... + a n .
a 1 ,
S n = a 1 +(a 1 + d)+(a 1 +2d)+ … +a n .
a n

S n = a n +(a n – d)+(a n –2d)+ … +a 1 .
S n = a 1 + (a 1 + d) + (a 1 +2d) + … + a n
+S n = a n + (a n − d) + (a n −2d) + … + a 1
_________________________________________________________
2S n = (a 1 + a n ) + (a 1 + a n ) + (a 1 + a n ) + … + (a 1 + a n )
(a 1 + a n ) n(a 1 + a n ).
S n .
2S n = n(a 1 + a n )
S n = n 2 (a 1 + a n)
S n ,
a 1 a n
S n = n 2 (a 1 + a n)
S n = n 2 (a 1 + a n). a 1 =8 d =5 n = 30,
a n
a n = a 1 + (n −1)d
Find a n where a 1 =8,d = 5 and n = 30. a 30 =8+(30 − 1)5
a 30 =8+(29)5
a 30 = 153
Knowing a 1 =8,n = 30, and a 30 = 153,
S n = n 2 (a 1 + a n)
S 30 = 30 (8 + 153)
2
S 30 = 15(161)
S 30 = 2,415

a n =3n −4.
S n = n 2 (a 1 + a n). n = 50,
a 1 a n
a 1 , n =1.
a n n = 50.
n = 50, a 1 =−1,
a 50 = 146
a n = 2n − 5.
a n =4n + 3.
25
∑ (4i +7).
i =1
S n = n 2 (a 1 + a n). n = 25,
a 1 a n

a 1 .
a 25 .
n = 25, a 1 = 11, a 25 = 107
30
∑ (6i − 4).
i = 1
35
∑ (5i − 3).
i = 1

4, 12, 20, 28, 36, 44, …
−7, −2, 3, 8, 13, 18, …
−15, −16, 3, 12, 21, 30, …
8, 5, 2, −1, −4, −7, …
11, 5, −1, −7 − 13, −19, …
15, 5, −5, −15, −25, −35, …
a 1 =11 d =7 a 1 =18 d =9
a 1 =−7 d =4 a 1 =−8 d =5
a 1 =14 d =−9 a 1 =−3 d =−3
−14
−19
−6.
−3.
−4 −2.
−8
−5.
−1 −4.
−11 −3.
−51.
−38
−6
−13

11, 14, 17, 20, 23, …
12, 18, 24, 30, 36, …
8, 5, 2, −1, −4, …
−17, −15, −13, −11, −9, …
16, 10, 4, −2, −8, …
−15, −12, −9, −6, −3, …
a n =5n −1 a n =2n +7
a n =−3n +5 a n =−4n +3
40
45
∑ (8i −7) ∑ (7i −5)
i =1
i =1
50
25
∑ (3i +6) ∑ (4i +3)
i =1
i =1
35
∑ (−6i −2)
i =1
30
∑ (−5i +1)
i =1
n
a
b

n
24
32 .
⎛ ⎞ 4 ⎠ .
a 3 4 b ⎛ 1
⎝ 2
f (x) = 4 · 3 x , a f (1) b f (2) c f (3).
a
n
a ,
n −1
a 4, 8, 16, 32, 64, 128, …

−2, 6, −12, 36, −72, 216, …
c 27, 9, 3, 1, 1 3 , 1 9 ,…
a
4, 8, 16, 32, 64, 128, …
8
4
16
8
32
16
64
32
128
64
2 2 2 2 2
r =2.
b
−2, 6, −12, 36, −72, 216, …
6
−2
−12
6
36
−12
−72
36
216
−72
−3 −2 −3 −2 −3
c
27, 9, 3, 1, 1 3 , 1 9 ,…
9
27 3 9 1 3
1
3
1
1
9
1
3
1
3
1
3 1 3 1 3 1 3
r = 1 3 .
a 7, 21, 63, 189, 567, 1,701, …
b 64, 16, 4, 1, 1 4 , 1
16
, …
c 2, 4, 12, 48, 240, 1,440, …

a −150, −30, −15, −5, − 5 2 , 0, …
b 5, 10, 20, 40, 80, 160, …
c 8, 4, 2, 1, 1 2 , 1 4 , …
a 1 ,
r =−2.
a 1 a 2 a 3 a 4 a 5
3 3·(−2) −6 · (−2) 12 · (−2) −24 · (−2)
−6 12 −24 48
3, −6, 12, −24, 48, …
r = −3.
r = −4.
a 1
a 1 .
a 1 ,
a 1 a 1
r · r r 2 a 1 r · r · r r 3
a 1 a 1
a n = a 1 r n −1
a 1

a n = a 1 r n −1
r = 1 2 .
a 14 ,
a 1 =64 r = 1 2 .
a n = a 1 r n −1
a 14 =64 ⎛ ⎝ 1 14 − 1
⎞
2⎠
a 14 =64 ⎛ ⎝ 1 13
⎞
2⎠
a 14 = 1
128
r = 1 3 .
r = 1 4 .
a n = a 1 r n −1 ,
a 1

a 1 =3
3, 6, 12, 24, 48, 96, …
6
3 12 6 24
12 48
24 96
48
2 2 2 2 2
r =2.
a 12 ,
a 1 = 3 and r =2.
a n = a 1 r n −1
a 12 =3·2 12 − 1
a 12 =3·2 11
a 12 = 6,144
a 1 = 3 and r =2.
a n = a 1 r n −1
a n =3(2) n −1
S n , S n = a 1 + a 2 + a 3 + ... + a n .
a 1 ,
S n = a 1 + a 1 r + a 1 r 2 + ... + a 1 r n −1
rS n = a 1 r + a 1 r 2 + a 1 r 3 + ... + a 1 r n

S n = a 1 + a 1 r + a 1 r 2 + a 1 r 3 +…+a 1 r n −1
rS n = a 1 r + a 1 r 2 + a 1 r 3 +…+a 1 r n −1 + a 1 r n
____________________________________________________
S n − rS n = a 1 −a 1 r n
S n (1−r) = a 1 (1−r n )
S n ,
(1 − r).
S n = a 1 (1−rn )
1−r
S n ,
S n = a 1 (1−rn )
1−r
a 1
S n = a 1 (1−rn )
. a
1−r
1 =7, r =2, n = 20.
a 1 =7,r =2, n = 20,
S n = a 1 (1−rn )
1−r
S 20 = 7⎛ ⎝1−2 20⎞ ⎠
1−2
S 20 = 7,340,025
k
∑ a(r) i ,
i =1

15
∑ 2(3) i .
i =1
S n = a 1 (1−rn )
, a
1−r
1
a 1
a 1 =6, r =3, n = 15,
15
∑ 6(2) i .
i = 1
10
∑ 5(2) i .
i = 1
a 1
a 1 + a 1 r + a 1 r 2 +…+a 1 r n −1 +…
a 1
a 1 + a 1 r + a 1 r 2 +…+a 1 r n −1 +…
S n = a 1 (1−rn )
.
1−r
3+6+12+24+48+96+ ….
a 1 =3 r =2

S n = a 1 (1−rn )
1−r
S n = a 1 (1−rn )
1−r
S n = a 1 (1−rn )
1−r
S 10 = 3⎛ ⎝1−2 10⎞ ⎠
S
1−2
30 = 3⎛ ⎝1−2 30⎞ ⎠
S
1−2
50 = 3⎛ ⎝1−2 50⎞ ⎠
1−2
S 10 = 3,069 S 30 = 3,221,225,469 S 50 ≈ 3.38 × 10 15
|r| ≥1
|r| ≥1.
1
2 + 1 4 + 1 8 + 16 1 + 32 1 + 64 1 +…
a 1 = 1 2 r = 1 2 .
S n = a 1 (1−rn )
1−r
S n = a 1 (1−rn )
1−r
S n = a 1 (1−rn )
1−r
S 10 =
⎛
1
2⎝ 1−⎛ ⎝ 1 ⎞
2⎠10⎞ ⎠
1− 1 2
S 20 =
⎛
1
2⎝ 1−⎛ ⎝ 1 ⎞
2⎠20⎞ ⎠
1− 1 2
S 30 =
⎛
1
2⎝ 1−⎛ ⎝ 1 ⎞
2⎠30⎞ ⎠
1− 1 2
S 10 ≈ .9990234375 S 20 ≈ 0.9999990463 S 30 ≈ 0.9999999991
|r|

= 18
54
r = 6 18 …
r = 1 3
a 1 . a 1 =54
r = 1 3
|r|

0. 5 – 0.5555555555555…
0.5 + 0.05 + 0.005 + 0.0005 + ...
r = 0.05
0.5
r = 0.005
0.05 …
a 1 . a 1 = 0.5
r = 0.1 r = 0.1 |r|

S =
a 1
1−r
a 1 = 1,000 r = 0.8.
S = 1,000
1 − 0.8
S = 5,000
A = P ⎛ ⎝1+ n r ⎞ nt
⎠
n =1.
A = P ⎛ ⎝1+ n r ⎞ nt
⎠
n =1. A = P ⎛ ⎝ 1+r 1t
1⎞
⎠
A = P(1+r) t
P(1+r) 1
P(1+r) 2 A = P(1+r) t
P(1+r) 1
P(1+r) 2
P(1+r) 1

1+r.
1+r
S t = a 1 (1−rt )
1−r
S t = P⎛ ⎝1−(1+r) t⎞ ⎠
1−(1+r)
S t = P⎛ ⎝1−(1+r) t⎞ ⎠
−r
S t = P⎛ ⎝(1+r) t −1 ⎞ ⎠
r
n
A t = P⎛ ⎝ ⎛ ⎝1+ r n ⎞ ⎠ nt −1 ⎞ ⎠
r
n
A t = P⎛ ⎝ ⎛ ⎝1+ n r ⎞nt ⎠ −1 ⎞ ⎠
r ,
n

P = 100
r = 0.05
n =12
t =18
P = 100, r = 0.05, n = 12 and
t = 18,
A t = P⎛ ⎝ ⎛ ⎝1+ r n ⎞ ⎠ nt −1 ⎞ ⎠
r
n
A t =
100 ⎛ ⎝ ⎛ ⎝1+ 0.05 · 18
⎞ ⎞
12 ⎠12
−1
⎠
0.05
12
A t = 34,920.20

3, 12, 48, 192, 768, 3072, …
2, 10, 50, 250, 1250, 6250, …
48, 24, 12, 6, 3, 3 2 ,…
54, 18, 6, 2, 2 3 , 2 9 ,…
−3, 6, −12, 24, −48, 96, …
2, −6, 18, −54, 162, −486, …
48, 24, 12, 6, 3, 3 2 ,…
12, 6, 0, −6, −12, −18, …
−7, −2, 3, 8, 13, 18, …
1
2
, 1 4 , 1 8 , 1
16
, 1
32
, 1
64
,…
5, 9, 13, 17, 21,25, …
4, 8, 12, 24, 48, 96, …
a 1 =4 r =3 a 1 =9 r =2
a 1 =−4 r =−2 a 1 =−5 r =−3
a 1 =27 r = 1 3
a 1 =64 r = 1 4
a 11 a 1 =8 r =3. a 13 a 1 =7 r =2.
a 10 a 1 =−6 r =−2.
a 15 a 1 =−4 r =−3.
a 10 a 1 = 100,000 r = 0.1. a 8 a 1 = 1,000,000 r = 0.01.
a 9 a 12
9, 18, 36, 72, 144, 288, …
5, 15, 45, 135, 405, 1215, …
a 15
−486, 162, −54, 18, −6, 2, …
a 16
224, −112, 56, −28, 14, −7, …
a 10
1, 0.1, 0.01, 0.001, 0.0001, 0.00001, …
a 9
1000, 100, 10, 1, 0.1, 0.01, …

8, 24, 72, 216, 648, 1944, …
7, 14, 28, 56, 112, 224, …
−6, 12, −24, 48, −96, 192, …
−4, 12, −36, 108, −324, 972, …
81, 27, 9, 3, 1, 1 3 ,…
256, 64, 16, 4, 1, 1 4 , 1
16
,…
15
∑ (2) i 10
∑ (3) i
i =1
i =1
9
∑ 4(2) i 8
∑ 5(3) i
i =1
i =1
10
∑ 9 ⎛ ⎝ 1 i
⎞
3⎠
i =1
15
∑ 4 ⎛ ⎝ 1 i
⎞
2⎠
i =1
1+ 1 3 + 1 9 + 27 1 + 81 1 + 243 1 + 729 1 +… 1+1 2 + 1 4 + 1 8 + 16 1 + 32 1 + 64 1 +…
6−2+ 2 3 − 2 9 + 2 27 − 2 81 +…
−4+2−1+1 2 − 1 4 + 1 8 −…
6 + 12 + 24 + 48 + 96 + 192 + …
1,024 + 512 + 256 + 128 + 64 + 32 + …
5 + 15 + 45 + 135 + 405 + 1215 + …
6,561 + 2187 + 729 + 243 + 81 + 27 + …
0. 3 – 0. 6 –
0. 7 – 0. 2 –
—
0. 45
—
0. 27

$100
d $1,500 80%
6%
a
$1,000 85%
b
$1,000 75%
c
$1,500 90%
$500
7%
$500
5%
$2,500
6.5%

a
b

7 · 6 · 5 · 4
4 · 3 · 2 · 1 .
(3x + 5) 2 .
(x − y) 2 .
(a + b) 2 .
(a + b) 2 = a 2 +2ab + b 2
(a + b) 3 , (a + b) 2 (a + b)
(a + b) 3
(a + b) 2 (a + b)
⎛
⎝a 2 +2ab + b 2⎞ ⎠(a + b)
a 3 +2a 2 b + ab 2 + a 2 b +2ab 2 + b 3
a 3 +3a 2 b +3ab 2 + b 3
(a + b) 3 = a 3 +3a 2 b +3ab 2 + b 3
(a + b) 7 ,

(a + b) 1 = a + b a 1 b 1
(a + b) 2 = a 2 +2ab + b 2 a 2 b 2
(a + b) 3 = a 3 +3a 2 b +3ab 2 + b 3 a 3 b 3
(a + b) 4 = a 4 +4a 3 b +6a 2 b 2 +4ab 3 + b 4 a 4 b 4
(a + b) 5 = a 5 +5a 4 b +10a 3 b 2 +10a 2 b 3 +5ab 4 + b 5 a 5 b 5
(a + b) n a n b n
a 0 =1,
(a + b) 3
(a + b) n
n +1.
a n b n .
(a + b) n ,
(a + b) n = a n + ___a n −1 b 1 + ___a n −2 b 2 + ... + ___a 1 b n −1 + b n .

(x + y) 6 .
a = x b = y.
(a + b) n = a n + ___a n −1 b 1 + ___a n −2 b 2 + ... + ___a 1 b n −1 + b n
(x + y) 6 = x 6 + ___x 5 y 1 + ___x 4 y 2 + ___x 3 y 3 + ___x 2 y 4 + ___x 1 y 5 + y 6

(x + y) 5 .
(p + q) 7 .
(x +3) 5 .
a = x b =3.
(a + b) n = a n + ___a n −1 b 1 + ___a n −2 b 2 + ... + ___a 1 b n −1 + b n
(x +3) 5 = x 5 + ___x 4 ·3 1 + ___x 3 ·3 2 + ___x 2 ·3 3 + ___x 1 ·3 4 +3 5
(x + 2) 4 .
(x + 1) 6 .
(3x −2) 4 .

a =3x b =−2.
(a + b) n = a n + ___a n −1 b 1 + ___a n −2 b 2 + ... + ___a 1 b n −1 + b n
(3x −2) 4 =1·( 3 +4(3x) 3 (−2) 1 +6(3x) 2 (−2) 2 +4(3x) 1 (−2) 3 +1·(−2) 4
(3x −2) 4 =81x 4 +4 ⎛ ⎝27x 3⎞ ⎠(−2) +6 ⎛ ⎝9x 2⎞ ⎠(4) +4(3x)(−8) +1·16
(3x −2) 4 =81x 4 − 216x 3 + 216x 2 −96x +16
(2x − 3) 4 .
(2x − 1) 6 .
⎛ n
⎝r ⎞ ⎠
⎛ n
⎝r ⎞ ⎠
n
r
⎛ n
⎝r ⎞ ⎠, 0≤r ≤ n,
⎛ n
⎝r ⎞ ⎠
n
r ⎞ ⎠ = n!
r !(n − r)!
⎛
⎝
a ⎛ ⎝ 5 ⎞
1⎠ b ⎛7 ⎞
⎝7⎠ c ⎛4 ⎞
⎝0⎠ d ⎛8 ⎞
⎝5⎠ .
a
n
r ⎞ ⎠ = n!
r !(n − r)! .
⎛
⎝

⎛ n
⎝r ⎞ ⎠ = n!
r !(n − r)! ,
n =5,r =1.
5!as5·4!
⎛
⎝ 5 ⎞
1⎠
5!
1!(5−1)!
5!
1!(4)!
5·4!
1!·4!
5·4!
1·4!
5
⎛
⎝ 5 ⎞
1⎠ =5
b
⎛ 7 ⎞
⎝7⎠
⎛ n
⎝r ⎞ ⎠ = n!
r !(n − r)! ,
n =7,r =7.
7!
7!(7−7)!
7!
7!(0)!
0!=1. 1
⎛ 7 ⎞
⎝7⎠ =1
c
⎛ 4 ⎞
⎝0⎠
⎛ n
⎝r ⎞ ⎠ = n!
r !(n − r)! ,
n =4,r =0.
4!
0!(4−0)!
4!
0!(4)!
1
⎛ 4 ⎞
⎝0⎠ =1

d
⎛ n
⎝r ⎞ ⎠ = n!
r !(n − r)! ,
n =8,r =5.
⎛ 8 ⎞
⎝5⎠
8!
5!(8−5)!
8!
5!(3)!
8!as8·7·6·5!
8·7·6·5!
5! ·3·2·1
56
⎛ 8 ⎞
⎝5⎠ =56
a ⎛6
⎝1⎠
⎞ b ⎛8
⎝8⎠
⎞ c ⎛5
⎝0⎠
⎞ d ⎛7
⎞
⎝3⎠
.
a ⎛2
⎝1⎠
⎞ b ⎛11
⎝11⎠
⎞ c ⎛9
⎝0⎠
⎞ d ⎛6
⎞
⎝5⎠
.
⎛n⎞
⎝1⎠ = n ⎛n⎞ ⎝n⎠ =1
⎛n⎞
⎝0⎠ =1
(a + b) n = ⎛ n⎞
⎝0⎠ an + ⎛ n⎞
⎝1⎠ an −1 b 1 + ⎛ n⎞
⎝2⎠ an −2 b 2 + ... + ⎛ n
⎝r ⎞ ⎠a n − r b r + ... + ⎛ n
⎝n ⎞ ⎠b n
(p + q) 4 .
a = p b = q.

(a + b) n = ⎛ n⎞
⎝0⎠ an + ⎛ n⎞
⎝1⎠ an −1 b 1 + ⎛ n⎞
⎝2⎠ an −2 b 2 + ... + ⎛ n
⎝r ⎞ ⎠a n − r b r + ... + ⎛ n
⎝n ⎞ ⎠b n
a = p, b = q n =4.
(p + q) 4 = ⎛4 ⎞
⎝0⎠ p4 + ⎛4 ⎞
⎝1⎠ p4−1 q 1 + ⎛4 ⎞
⎝2⎠ p4−2 q 2 + ⎛4 ⎞
⎝3⎠ p4−3 q 3 + ⎛4 ⎞
⎝4⎠ q4
⎛n⎞
⎝1⎠ = n, ⎛n⎞ ⎝n⎠ =1,
(p + q) 4 = ⎛4 ⎞
⎝0⎠ p4 + ⎛4 ⎞
⎝1⎠ p3 q + ⎛4 ⎞
⎝2⎠ p2 q 2 + ⎛4 ⎞
⎝3⎠ pq3 + ⎛4 ⎞
⎝4⎠ q4
⎛n⎞
⎝0⎠ =1.
(p + q) 4 =1p 4 +4p 3 q 1 + 4!
2!(2)! p2 q 2 + 4!
3!(4−3)! p1 q 3 +1q 4
(p + q) 4 = p 4 +4p 3 q +6p 2 q 2 +4pq 3 + q 4
(x + y) 5 .
(m + n) 6 .
(p + q) 4
(x −2) 5 .
a = x b =−2.
(a + b) n = ⎛ n⎞
⎝0⎠ an + ⎛ n⎞
⎝1⎠ an −1 b 1 + ⎛ n⎞
⎝2⎠ an −2 b 2 + ... + ⎛ n
⎝r ⎞ ⎠a n − r b r + ... + ⎛ n
⎝n ⎞ ⎠b n
a = x, b =−2, n =5.

(x −2) 5 = ⎛ ⎝ 5 ⎞
0⎠ x5 + ⎛ ⎝ 5 ⎞
1⎠ x5−1 (−2) 1 + ⎛ ⎝ 5 ⎞
2⎠ x5−2 (−2) 2 + ⎛ ⎝ 5 ⎞
3⎠ x5−3 (−2) 3 + ⎛ ⎝ 5 ⎞
4⎠ x5−4 (−2) 4 + ⎛ ⎝ 5 ⎞
5⎠ (−2)5
⎛ n⎞
⎝1⎠ = n, ⎛n⎞ ⎝n⎠ =1,
⎛n⎞
⎝0⎠ =1.
(x −2) 5 = ⎛ ⎝ 5 ⎞
0⎠ x5 + ⎛ ⎝ 5 ⎞
1⎠ x4 (−2) + ⎛ ⎝ 5 ⎞
2⎠ x3 (−2) 2 + ⎛ ⎝ 5 ⎞
3⎠ x2 (−2) 3 + ⎛ ⎝ 5 ⎞
4⎠ x(−2)4 + ⎛ ⎝ 5 ⎞
5⎠ (−2)5
(x −2) 5 =1x 5 +5(−2)x 4 +
2!·3! 5! (−2)2 x 3 +
3!·2! 5! (−2)3 x 2 +
4!·1! 5! (−2)4 x +1(−2) 5
(x −2) 5 = x 5 +5(−2)x 4 +10·4·x 3 +10(−8)x 2 +5·16·x +1(−32)
(x −2) 5 = x 5 −10x 4 +40x 3 −80x 2 +80x −32
(x − 3) 5 .
⎛
⎝y
− 1 ⎞6 ⎠ .
⎛ ⎝2x −3y ⎞ ⎠4 .
a =2x b =−3y.
(a + b) n = ⎛ n⎞
⎝0⎠ an + ⎛ n⎞
⎝1⎠ an −1 b 1 + ⎛ n⎞
⎝2⎠ an −2 b 2 + ... + ⎛ n
⎝r ⎞ ⎠a n − r b r + ... + ⎛ n
⎝n ⎞ ⎠b n
a =2x, b =−3y n =4.
(2x −3y) 4 = ⎛4 ⎞
⎝0⎠ (2x)4 + ⎛4 ⎞
⎝1⎠ ⎝−3y (2x)4−1⎛ ⎞ ⎠1 + ⎛4 ⎞
⎝2⎠ ⎝−3y (2x)4−2⎛ ⎞ ⎠2 + ⎛4 ⎞
⎝3⎠ ⎝−3y (2x)4−3⎛ ⎞ ⎠3 + ⎛4 ⎞
⎝4⎠ ⎛ ⎝−3y ⎞ 4
⎠
(2x −3y) 4 = ⎛4 ⎞
⎝0⎠ (2x)4 + ⎛4 ⎞
⎝1⎠ (2x)3 ⎛ ⎝−3y ⎞ ⎠1 + ⎛4 ⎞
⎝2⎠ (2x)2 ⎛ ⎝−3y ⎞ ⎠2 + ⎛4 ⎞
⎝3⎠ (2x)1 ⎛ ⎝−3y ⎞ ⎠3 + ⎛4 ⎞
⎝4⎠ ⎛ ⎝−3y ⎞ 4
⎠
⎛n⎞
⎝1⎠ = n, ⎛n⎞ ⎝n⎠ =1,
⎛n⎞
⎝0⎠ =1.
(2x −3y) 4 =1(2x) 4 +4(2x) 3 ⎛ ⎝−3y ⎞1 ⎠ +
2!(2)! 4! (2x)2 ⎛ ⎝−3y ⎞2 ⎠ + 4!
3!(4−3)! (2x)1 ⎛ ⎝−3y ⎞3 ⎠ +1 ⎛ ⎝−3y ⎞ ⎠4
(2x −3y) 4 =16x 4 +4·8x 3 ⎛ ⎝−3y ⎞ ⎠ +6 ⎛ ⎝4x 2⎞ ⎠ ⎛ ⎝9y 2⎞ ⎠ +4(2x) ⎛ ⎝−27y 3⎞ ⎠ +81y 4
(2x −3y) 4 =16x 4 −96x 3 y + 216x 2 y 2 − 216xy 3 +81y 4
⎛ ⎝ ⎛ 3x − 2y ⎞ ⎠ ⎞5 .
⎛ ⎝ ⎛ 4x − 3y ⎞ ⎠ ⎞4 .

(r +1)st
(r +1)st
(r +1) st (a + b) n
n
r ⎞ ⎠a n − r b r
⎛
⎝
(x + y) 7 .
n =7, a = x b = y.
Since r +1 = 4,
then r = 3.
n =7, r =3, a = x, b = y.
(x + y) 6 .
(a + b) 8 .
x 6 (x +3) 9 .

n =9, a = x, b =3.
x 6
a = x, x 9−r = x 6 , r =3.
n =9, r =3, a = x, b =3.
x 6
x 5 (x + 4) 8 .
x 4 (x + 2) 7 .

(x + y) 4 (a + b) 8
(m + n) 10 (p + q) 9
(x − y) 5 (a − b) 6
(x +4) 4 (x +5) 3
⎛ ⎝y +2 ⎞ 5
⎠
⎛ ⎝y +1 ⎞ 7
⎠
(z −3) 5 (z −2) 6
(4x −1) 3 (3x −1) 5
(3x −4) 4 (3x −5) 3
⎛ ⎝2x +3y ⎞ 3
⎠
⎛ ⎝3x +5y ⎞ 3
⎠
a ⎛8 ⎞
⎝1⎠ b ⎛10
⎞
⎝10⎠ c ⎛ ⎝ 6 ⎞
0⎠ d ⎛9 ⎞
⎝3⎠
a ⎛3 ⎞
⎝1⎠ b ⎛9 ⎞
⎝9⎠ c ⎛7 ⎞
⎝0⎠ d ⎛ ⎝ 5 ⎞
3⎠
a ⎛7 ⎞
⎝1⎠ b ⎛4 ⎞
⎝4⎠ c ⎛3 ⎞
⎝0⎠ d ⎛10
⎞
⎝8
⎠
a ⎛4 ⎞
⎝1⎠ b ⎛ ⎝ 5 ⎞
5⎠ c ⎛8 ⎞
⎝0⎠ d ⎛11
⎞
⎝9
⎠
(x + y) 3 (m + n) 5
(a + b) 6 (s + t) 7
(x −2) 4
⎛ ⎝p −1 ⎞ 5
⎠
⎛ ⎝3x − y ⎞ 5
⎠
⎛ ⎝2x +5y ⎞ 4
⎠
⎛ ⎝y −3 ⎞ 4
⎠
⎛ ⎝q −4 ⎞ 3
⎠
⎛ ⎝5x −2y ⎞ 4
⎠
⎛ ⎝3x +4y ⎞ 5
⎠

(x + y) 10 (a + b) 9
(x − y) 8 (x − y) 11
y 3 ⎛ ⎝y +5 ⎞ 4
⎠
x 6 (x +2) 8
x 5 (x −4) 6 x 7 (x −3) 9
a 4 b 2 (2a + b) 6
p 5 q 4 ⎛ ⎝3p + q ⎞ 9
⎠
(a + b) n (a − b) n .
a
b

a n − a n −1 ,
a n
a , n −1
a n ,
n!
0! 0!=1
n! =n(n −1)(n −2) … (3)(2)(1).
a n
n
∑ a i = a 1 + a 2 + a 3 + a 4 + a 5 +…+a n
i =1
a 1
a n = a 1 + (n −1)d
S n , a 1 a n
S n = n 2 (a 1 + a n)
a 1

a n = a 1 r n −1
S n ,
a 1
S n = a 1 (1−rn )
1−r
a 1
a 1 + a 1 r + a 1 r 2 +…+a 1 r n −1 +…
a 1
If |r| < 1, the sum is
S = a 1
1−r
We say the series converges.
If |r| ≥ 1, the infinite geometric series does not have a sum. We say the series diverges.
n
(a+b) n
n +1.
a n b n .
A t = P⎛ ⎝ ⎛ ⎝1+ r n ⎞ ⎠ nt −1 ⎞ ⎠
r
n
n
r ⎛ ⎝
⎛
⎝
n
r ⎞ ⎠, 0≤r ≤ n,
n
r ⎞ ⎠ = n!
r !(n − r)!
⎛ n
⎝r ⎞ ⎠

⎛n⎞
⎝1⎠ = n ⎛n⎞ ⎛n⎞
⎝n⎠ =1 ⎝0⎠ =1
(a + b) n = ⎛ n⎞
⎝0⎠ an + ⎛ n⎞
⎝1⎠ an −1 b 1 + ⎛ n⎞
⎝2⎠ an −2 b 2 + ... + ⎛ n
⎝r ⎞ ⎠a n − r b r + ... + ⎛ n
⎝n ⎞ ⎠b n
a n =7n −5 a n =3 n +4
a n =2 n + n a n =
2n +1
4 n
a n = (−1)n
n 2
9, 18, 27, 36, 45, … −5, −4, −3, −2, −1, …
1
e 3, 1
e 2, 1 e ,1,e, … 1, −8, 27, −64, 125, …
− 1 3 , − 1 2 , − 3 5 , − 2 3 , − 5 7 ,…
a n =4n! a n = n!
(n +2)!
a n =
(n −1)!
(n +1) 2
7
3
∑ (2i −5) ∑ 5 i
i =1
i =1
4
∑ 4
k !
k =0
4
∑ (k +1)(2k +1)
k =1

− 1 3 + 1 9 − 27 1 + 81 1 − 243
1
4 − 8 + 12 − 16 + 20 − 24
4+2+ 4 3 +1+4 5
1, 2, 4, 8, 16, 32, … −7, −1, 5, 11, 17, 23, …
13, 9, 5, 1, −3, −7, …
a 1 =5 d =3 a 1 =8 d =−2
a 1 = −13 d =6
−5.
−21
12
14
−3.
−26
−91.
7, 4, 1, −2, −5, … 1, 6, 11, 16, 21, …
a n =4n +7 a n =−2n +19

50
30
∑ (4i −5) ∑ (−3i −7)
i =1
i =1
35
∑ (i + 10)
i =1
3, 12, 48, 192, 768, 3072, … 5, 10, 15, 20, 25, 30, …
112, 56, 28, 14, 7, 7 2 ,… 9, −18, 36, −72, 144, −288, …
a 1 =−3 r =5 a 1 = 128 r = 1 4
a 1 =5 r =−3
a 9 a 1 =6 r =2. a 11 a 1 = 10,000,000 r = 0.1.
a 12 a 9
6, −24, 96, −384, 1536, −6144, …
4374, 1458, 486, 162, 54, 18, …
−4, 8, −16, 32, −64, 128… 3, 12, 48, 192, 768, 3072…
3125, 625, 125, 25, 5, 1…
8
∑ 7(3) i 6
∑ 24 ⎛ ⎝ 1 i
⎞
2⎠
i =1
i =1
1− 1 3 + 1 9 − 27 1 + 81 1 − 243 1 + 729 1 −… 49+7+1+1 7 + 49 1 + 343 1 +…
0. – —
8 0. 36

$360
60%
(a + b) 7 (x − y) 4
$300
7%
(x +6) 3
⎛
⎝2y −3 ⎞ 5
⎠
⎛
⎝7x +2y ⎞ 3
⎠
a ⎛11
⎞
⎝1
⎠
b ⎛12
⎞
⎝12⎠
c ⎛13
⎞
⎝0
⎠
d ⎛8 ⎞
⎝3⎠
a ⎛7 ⎞
⎝1⎠
b ⎛ ⎝ 5 ⎞
5⎠
c ⎛9 ⎞
⎝0⎠
d ⎛9 ⎞
⎝5⎠
a ⎛1 ⎞
⎝1⎠
b ⎛ ⎝ ⎞
15⎠
c ⎛4 ⎞
⎝0⎠
d ⎛11
⎞
⎝2
⎠
(p + q) 6 (t −1) 9
(2x +1) 4
⎛
⎝4x +3y ⎞ 4
⎠
⎛
⎝x −3y ⎞ 5
⎠

(a + b) 9 (x − y) 7
y 4 ⎛ ⎝y +3 ⎞ 6
⎠
x 5 (x −2) 8
a 3 b 4 (2a + b) 7

a n = 5n −3
(n +2)!
3 n a n =
(n +3)!
− 2 3 , − 4 5 , − 6 7 , − 8 9 , − 10
11 ,… 4
(−4) i
∑
i =1
−1 + 1 4 − 1 9 + 16 1 − 25
1
a 1 = −13 d =3
−7.
−1
−15.
11
5, 9, 13, 17, 21, …
a n =−3n + 100.
40
∑ (5i − 21)
i =1
14, 3, −8, −19, −30, −41, … 324, 108, 36, 12, 4, 4 3 ,…
a 1 =6 r =−2
a 1 =5 r =4, a 11 .
a 10
1250, 250, 50, 10, 2, 2 ,….
5
2, −6, 18, −54, 162, −486…
9
∑
i =1
5(2) i 1− 1 5 + 1 25 − 1
125 + 1
625 − 1
3125 +…
—
0. 81

(m −2n) 5 a ⎛8 ⎞
⎝1⎠
b ⎛ ⎝ ⎞
16⎠ c ⎛12
⎞
⎝0
⎠ d ⎛10
⎞
⎝6
⎠
⎛ ⎝4x +5y ⎞ 3
⎠

a b c d
a b c d
2·2·2·2·5
e
e
2·2·3·5 36 72
a b c a b c
10x 2 +16x +17 12y 2 +9y +7 a 14x 2 −13 b 12x ÷2
a 17y 2 +19 b 7y
c x +11 d 11a −14
c z +13 d 8x −18
a 4(p + q) b 4p + q a 2x −8 b 2(x −8)
w −7 l −6 4q −8
7n +3 a = b > c <
d =
a > b = c >
d <
a −6 b c −2
a −7 b c −3 a b −2 c −10 d a b −3 c −11 d
a 8, 8 b −18,
a 8, 8 b −22,
c 19, 19 d −4,
−4
−18
c 23, 23 d 3, 3
a b c −63 d −9 a b c d
a b −81 a b −49 a b
a b
⎛ ⎝9+(−16) ⎞ ⎠ +4; −3 ⎛ ⎝−8 + (−12) ⎞ ⎠ +7; −13
− 23
40 − 5 8
−33a
−26b
4
15
2
3
3
4b
4
q
79
60
103
60 a
a
24k −5
30
b
15
2k
1
90
2
7 − 1 2
2
3
a 6.58 b 6.6 c a 15.218 b 15.22
c 15.2
−22
27a −32
36
b 2a 3
a −16.49 b −0.42
a −23.593 b −12.58 −27.4815 −87.6148
a b c a b c 587.3
34.25
a 117
500 b −0.875 a 3
125 b −0.375

a b
a b
a b c −10 a 4, 49 b −3, 4, 49
a b c −15
a 6, 121
c −3, 0. 3 – , 9 5 ,4, 49 d −2 b −25, −1, 6, 121
e −3, − 2, 0.3 – , 9 5 ,4, 49 c −25, − 3 , −1, 6, 121
8
d 2.041975...
e
−25, − 3 , −1, 6, 121, 2.041975...
8
a
b
a
32r +29s 41m +6n
b
1 7 15 1 2 9
−92x
5
49
−48a
a b a b 4x +8
6x +42 5y +3 4n +9
70 + 15p 4+35d −10+15a
−56 + 105y −z +11 −x +4
3−3x 2x −20 5x −66
7x −13
2·43 5·7·13 2·2·2·2·3·3·3
a b c
10x +6 22a +1 17x 2 +20x +16
11
25

a 5x 2 −6xy b 6y2
5x
c y 2 +21 d 81x 2 −6x
a 4ab 2 +3a 2 b b 20xy 2
c m +15 d 121x 2 −9x
a 5 ⎛ ⎝3x + y ⎞ ⎠ b 15x + y 2c +14 3n −7
a 8 ⎛ ⎝y −9 ⎞ ⎠ b 8y −9
a > b > c > d >
a = b = c > d =
a −11 b −3 c 3
−22 29
a b −6 c −20 d 20 −59
a b a b
a −32 b −65 c −4
a −4 b −12 c −39
d 14
a 64 b −64
d 13
−47
90 9 41
−9 −29 1
−3 5 a −47 b 16
121 −56 6
⎛ ⎝3+(−15) ⎞ ⎠ +7; −5 a 10 − (−18); 28
−6
a + b
b −25 − 11; − 36
96° −$28
10
21
2x 2
3y
− 12 7
− 21a2
11b 2
11
30
1
3 − 21
50
9n
33
4x − 4 9
10u
9v −
16 1 − 10 9
− 2 5
2m
3n
29
24
1
48
17
105 − 53
40
1
12
4x +3
12 a 5 6 b 4
a 25n
16
b
25n −16
30 a
−8x −15
18
b − 10k
−5(a +1)
27 a
3
9
7
−8
11
6
5
2
54
49
25
15
4
5
21
5
4
1
24
7
9
−28−15y
60
−5
33
64
23
24
b a

11 1
5
13
3
a 1 5 b 6 5 − 1 9 − 5 11
a b c
a b c a b c −40.91
−7.22 −27.5 102.212
51.31 −4.89 −1200.47982
337.8914 1.305 92.4
$2.44
−4.8 2.08
1
25
19
200
0.85 −12.4
0.71 0.393 156%
6.25%
−10 a 0, 36, 9b −8, 0, 36, 9
a b − 100, −7,−1
c −8, 0, 12 5 , 36,9 d c − 100, −7, − 8 3 , −1, 0.77, 31 4
1.95286..., e
d
−8, 0, 1.95286..., 12 5 , 36,9
e
− 100, −7, − 8 3 , −1, 0.77, 31 4
27m + (−21n)
5
4 g + 1 2 h 2.43p + 8.26q
−63
1 5 6
17
14.88 10p 44
7
8
d
49
11
32y +72
6c −78
3
4 q +3
5y −3 3+8r 36d +90
rs −18r yp +4p −28p −7
−3x +18 −3x +7 −3y −8
−33c +26 −a +19 4m −10
72x −25 22n +9 6c +34
12y +63
2, 3, 5, 6, 10

6x 2 − x +5 a 11 ⎛ ⎝y −2 ⎞ ⎠ b 11y −2
a = b = c > d = −9 a b −8 c −22 d
a b
a −3 b −15 c −56 d
−12 ⎛ ⎝−4 + (−9) ⎞ ⎠ + 23; 10
46 °
8x
−15x3
11y 2
15y
31
36 a 11 8 b 5 6 − 1 6
− 1 5
6.17
96.978 56.3
−2.3
− 48 5
1.27
4.75%
3
4 x + y 1 11
15
8b +10 xp −5p
−6x −6 6y +16
7n +7 −8 − 11; − 19
⎛
⎝−8 − (−3) ⎞ ⎠ +5;0
−8
9
17
a 28.15 b 28.146
15
17 − 5 3
− 7 6
−65.4
1 8 13
a b
3
13y −3
18x +30 3−2n 12
2x −6 0.045 60%
16 w 2 −3w 63
x =6 2c +3 0.0042
12 x x −15
2
5 x +4
−x +4 −7
12

a b a b m =2
a =0 u =2 x =4
p =−2 q =−8
y =− 17 5
z =0
q =−
11
9 k = 201
14
x = 1 2
x =−2
x =12 u = −12 n =2
m =−1 r =1 s =−8
n =9 d =16
3 6
−15, −8 −29, 11 −33, −32, −31
−13, −12, −11 −10, −8, −6
a b c 125%
a b a 175%
8.8%
a b a b
b = 2A h h = 2A b C = 5 (F − 32)
9
b = 2A − Bh
h
t = A Pr
− P
r = A Pt
− P
y = 9−4x
7 y = 1−5x
8

1.8
a
a
b
b
c
c
a
a
a
b
b
b
c
c
c

a
a
a
b
b
b
c

a ±11 b c
a ±2 b c
x =4,x =− 2 3 x =−1,x = 5 2
x =8,x =0 x =8,x =2
x =− 2 5 , x = 5 2 x =3, x = 1 9

a b a b y =5
w = −18 q =3 x =14
c =4
c = 5 2
n =2
y =−5 s =10 p =−4
c =−4
h = 3 4
m =6
j = 2 5
m = 16 5
x =−1 y =−1
a = 3 4
x =4
w = 9 4
b =12
p = −41
x =− 5 2
n =−3
x =−2 u =3 x =18
x =20 n =9 d =8
L = 19.75
−5
−11, −12, −13
−69, −71, −73
a b a a 162.5 b a
$11.88
−2.5% −11%
a $26.97 b $17.98 a b a $7.20 b $23.20
a $0.20 b $0.80
3.75%
17.5%
d = C π L =
WH
V
b = 2A h
d 1 = 2A
d 2
b 1 = 2A h − b 2
a = 2h − 108t
t 2
a = 180 − b − c
p =
2A −2B
l

L = P −2W
2
y =4+x
y =15−8x
y =−6+4x
y = 7−4x
3 y = 12 − 2x
3
y = 18 − 3x
−2
45°, 45°, 90° 30°, 60°, 90°
10.2 9.8
13.5 12.8
104° F
a b
2×8
4×4
c
1.6
4.8
5.9%
a b
c
a
3.6 5.4
a
b
b
c
c

a
a
a
b
b
b
c
c
c
a
a
a
b
b
b
c

5≤n ≤24 6≤w ≤12 a b
a x =4,x =−4b
c z =0
a x =3,x =−3b
c z =0
x =1,x =− 1 2 x =−1,x =− 5 2
x =7,x =1
x =1,x =−5 x =7,x =1
x =−1,x =− 2 3
x =−3,x =3

x =2,x = 1 4
x = 3 2 , x =−1 2
x =3,x =2
x =3,x =− 11 3
s = −22 m = −14
q =18 x =−1
k = 3 4
n =2 k =23
x =5
−3, −10
$3.89
a b 52.5%
d 2 = 2A
d 1
y = 4x
3 −4
22.5°, 67.5°, 90°
24.5 12.5
2.2 7.8 9.7%

6≤w ≤12
x =9,x =−3
x =2,x = 1 4
x =2,x = 2 3
x =−5 a =41
x =6
x =−2,x =− 1 3

10.8 −57, −55
30°, 60°, 90° 2.5
−9 18 y =−4
− 1 2
0 −5; − 5; 5
2
5 x +6 −3x −6 y =−2x +1
x >5 x >5
−11 2a 2 − a −3 3x +4
a 8 b 9 a 7 b 3 a 2 b 4
a b

a b
a
b

a
b
(2, 0),
(0, −2)
(8, 0),
(0, 2)
(3, 0),
(0, 2)
(4, 0),
(0, 12)

− 4 3 − 3 5
−1
a m = 2 ; (0, −1)
5
b m =− 3 ; (0, 6)
2
b m =− 1 4 ; (0, 2) a m =− 4 3
; (0, 1)
a b
c d
a b
c
d

a
b
c
d
a
b
c
1
4 ,
d
a
b
c
d
a
b
c 1.8,
$1.80
d
a b
a b
a b a b a b
a b y = 2 5 x +4
y =−x −3
y = 3 5 x +1 y = 4 3 x −5 y =− 2 5 x −1
y =− 3 4 x −4 y =8 y =4

y = 1 3 x − 10 3 y =− 2 5 x − 23 5
x =5
x =−4
y =3x −10
y = 1 2 x +1
y =− 1 3 x + 10 3
y =−2x +16 y =−5
y =−1 x =−5 x =−4
a b c d a b c d
y ≥−2x +3
e
e
y ≤ 1 2 x −4 x −4y ≤8 3x − y ≥6
y > 5 2 x −4.
y < 2 3 x −5.
2x −3y 3.
y >−3x.
y ≥−2x.

a 10x +13y ≥ 260
b
y

a b c a f (3) =22b f (−1) =6
a (2) =13b f (−3) =3
a 4m 2 −7 b 4x −19
c 4x −12
a b
c f (t) =3t 2 −2t −1
a 2k 2 +1 b 2x +3
c 2x +4
a b
c f (h) =2h 2 +4h −3
a b
a b

⎡ ⎣−5, 1 ⎤ ⎦.
[−4, 2].
a f (0) = 1 b f (π) =−1
c f (−π) =−1d f (x) =0
e
x =− 3π 2 , − π 2 , π 2 , 3π 2
⎛
⎝ −3π 2 ,0⎞ ⎠ , ⎛ ⎝ −π 2 ,0⎞ ⎠ , ⎛ ⎝ π 2 ,0⎞ ⎠ , ⎛ ⎝ 3π 2 ,0⎞ ⎠
f (0, 1) g (−∞, ∞) h
[−1, 1]
[−2, 4].
⎡ ⎣−5, 3 ⎤ ⎦.
a f (0) = 0 b f = ⎛ ⎝ π ⎞
2⎠ =2
c f = ⎛ ⎝ −3π
2
⎞
⎠ =2d f (x) =0
x =−2π, −π, 0,π, 2π e
(−2π, 0), (−π, 0), (0, 0), (π, 0), (2π, 0)
f (0, 0) g (−∞, ∞) h
[−2, 2]
a b
a b

a
a
b
b

(3, 0), (0, 3) (5, 0), (0, −5)
(5, 0), (0, −5) (2, 0), (0, 6) (2, 0), (0, −8)
(5, 0), (0, 2)

2
5
5
4 − 1 3
− 5 2
− 5 2 − 8 7
7
3
−1
m =−7;(0, 3)
m =−3;(0, 5)
m =− 3 2 ; (0, 3) m = 5 ; (0, −3)
2

a
b
c 1.75,
$1.75
d

a
b
c
d
a
b
c 0.15,
d
a
b
c
d
y =3x +5
y =−3x −1
y = 1 5 x −5 y =−1
y =3x −5
y =−2
y = 1 2 x −3 y =− 4 3 x +3
y = 5 8 x −2 y =− 3 5 x +1
y =− 3 2 x −9 y =−7x −10 y =5
y =−7 y =−x +8
y = 1 4 x − 13 4
y =2x +5
y =− 7 2 x +4 x =7
y =−4 y =4x −2 y =2x −6
x =−3 y =−2
y = 1 2 x +1
y =− 4 3 x y =− 3 2 x +5 y = 5 2 x
y =4 y =−4 x =−2
y =4
y =− 1 2 x +5
y = 1 6 x
y =− 4 3 x −3 y =− 3 4 x +1 x =−2
x =−2
a b c d e
y =− 1 5 x − 23 5
a b c d e
y ≤3x −4
y =−2x −2
a b c d e
y ≤ 1 2 x +1

x + y ≥5 3x − y ≤6

a 11x + 16.5y ≥ 330
b
a 15x +10y ≥ 500
b
c
a b
a
b
c
a b
c
c
a b
a
b
c
a b
c
a
b
c
a
b c
a b
c

a b
a b c
a b c
c
a f (2) =7b f (−1) =−8
c f (a) =5a −3
a f (2) =9b f (−1) =6
c f (a) =2a 2 − a +3
a g(h 2 )=3−h 2
b g(x +2) =1−x
c g(x) + g(2) =4−x
a f (2) =−6b f (−1) =6
c f (a) =−4a +2
a g(h 2 )=2h 2 +1
b g(x +2) =2x +5
c g(x) + g(2) =2x +6
a f (2) =5b f (−1) =5
c f (a) =a 2 − a +3
a g(h 2 )=−3h 2 −2
b g(x +2) =−3x −8
c g(x) + g(2) =−3x −10
a
b N(4) = 165
a
a b
a b
b N(0) = 1500
c N(1000) = 4750
a
a
a
b
b
b

a
a
a
b
a
b
a
b {−3}
a
b
a
b
a
b
a
b
b
b −1,

a
a
a
b
b
a
b
a
a
b
b D: (−∞, ∞), R:[0, ∞) b
[−2, 2],
a f (0) =0b f ⎛ ⎝ π ⎞
2⎠ a 5 b 2 c 2 d f (x) =0
c f ⎛ ⎝ −3π ⎞
2 ⎠ =−1d f (x) =0
x =−2π, −π, 0,π, 2π
e (−2π, 0), (−π, 0),
(0, 0), (π, 0), (2π, 0)
f (0, 0) g (−∞, ∞)
h [−1, 1]
e f (0, 5) g
[−3, 3]
h ⎡ ⎣2, 5 ⎤ ⎦
b c

(0, 3)(3, 0)
(6, 0), (0, 3) (16, 0), (0, −12)
− 1 2

−6
5
2
m = 5 3 ; (0, −6) m = 4 5 ; ⎛ ⎝ 0, − 8 ⎞
5⎠

a −$250
b
c
d
y =−5x −3
y =−2x y =−3x +5 y =−4
y = 3 5 x
y =−2x −5
y = 1 2 x − 5 2
y =2
y =− 2 5 x +8 y =3
y =− 3 2 x −6 y =1 a b c d e
y ≥ 2 3 x −3 x −2y ≥6

a 20x +15y ≥ 600
b
a
b
a
a
b
c
c
a b
c
b
c
a f (−2) = −10 b f (3) =5
c f (a) =3a −4
a f (−2) =20b f (3) =0
c f (a) =a 2 −5a +6
a
b

a
a
a
b
b
a
b
a
b −2,
a f (x) =0b f ⎛ ⎝ π ⎞
2⎠ =1
c f ⎛ ⎝ −3π ⎞
2 ⎠ =1d f (x) =0
x =−2π, −π, 0,π, 2π
e (−2π, 0), (−π, 0), (0, 0),
(π, 0), (2π, 0) f (0, 0)
g [−∞, ∞] h [−1, 1]
b
a − 3 5 b

y =2x +5
y =− 4 5 x −5
a b
{−3,−2,−1,0,1,2,3} c
a
a x =−2,2b y =−4
c f (−1) =−3d f (1) =−3
e f −4,
b
a b a b (3, 2)
(2, 3) (3, 4) (5, −4)
a a
b
(6, 1) (−3, 5)
b
⎛ ⎝ 2, 3 ⎞
2⎠

⎛ ⎝ −1 2 ,−2⎞ ⎠
(2, −1) (−2, 3)
(1, 3) (4, −3) (6, 2)
(1, −2)
a a
b
b
−8
1 1 2

a C(x) =15x + 25, 500 b
R(x) =32x c
a C(x) = 120x + 150,000 b
R(x) = 170x c
d 1,500
a b a b (2, −1, 3)
d 3,000
(−2,3,4) (−3, 4, −2) (−2,3,−1)
(x, 3,z)
x = z −3;y =3;z
(x, y, z)
x =5z −2;y =4z −3;z
⎧ x − y +2z =3
a ⎡ ⎣ 3 8 ⎤
2 5 −3⎦
a ⎡ 11 9 −5⎤
⎨2x + y −2z =1
⎣7 5 −1⎦
⎩
b ⎡ −5 3 8⎤
⎢23 −1 4 7⎥
b ⎡ 4x − y +2z =0
−3 2 −5⎤
⎢52 −1 −1 4⎥
⎣1 3 2 −3⎦
⎣3 −2 2 −7⎦
⎧ x + y + z =4
⎨2x +3y − z =8
⎩ x + y − z =3 a ⎡ −2 3 0 −2
⎣ ⎢ ⎤
4 −1 −4 4⎥
a ⎡ 4 1 −3 2
5 −2 −2 −2⎦
⎣ ⎢ ⎤
2 −3 −2 −4⎥
5 0 4 −1⎦
b ⎡ −2 3 0 −2
⎣ ⎢ ⎤
4 −1 −4 4⎥
b ⎡ 2 −6 4⎤
⎢82 −3 −2 −4⎥
15 −6 −6 −6⎦
⎣5 0 4 −1⎦
c
c ⎡ 14 −7 −12 −8
⎡−2 3 0 −2 ⎤ ⎣ ⎢ ⎤
2 −3 −2 −4⎥
⎢ 3 4 −13 −16 −8⎥
5 0 4 −1⎦
⎣ 15 −6 −6 −6 ⎦
⎡ ⎣ 1 −1 2 ⎤
0 −3 −4⎦ ⎡ ⎣ 1 −1 3 ⎤
(4, −1).
0 −5 8⎦
(−2, 0). (6, −1, −3) (5,7,4)

(x, y, z),
x = z −3;y =3;z
(x, y, z),
a −14; b −28
a b −15
x =5z −2;y =4z −3;z
a b c a −3 b c
−11 −12
(− 15 7 , 24 7 ) (−2, 0) (−9,3,−1)
(−6, 3, −2)
a b
a b

a ⎧ +20p ≤ 160
⎨30m
⎩2m +3p ≤15
b
a ⎧ ≥ p +5
⎨a
⎩a +2p ≤ 400
b
a ⎧ ⎨0.75d
+2e ≤25
⎩360d + 110e ≥ 1000
b
c
d
a ⎧ + 125j ≥ 1000
⎨140p
⎩1.80p + 1.25j ≤12
b
c
d
c
d
c
d
a b a b (−2, 3)
(0, 2) (2, 4) (−2, −2)
(3, 3) (6, −4)
(1, −4) (−3, 2)
⎛ ⎝ −1 2 , 5 ⎞
2⎠
(−5, 4) (0, 10) (4, −2)
(4, 0) (4, 5)
(7, 12) (−3, −5) (2, −3)
(−11, 2) (6, − 9) (−3, 2)
a b

a b
−7 −19
−67
53.5 36.5
7.5
4.5%
5.5%
a C(x) =5x + 6500 b
R(x) =10x c
d
a b a b (4, 5, 2)
(7, 12, −2) (−3, −5, 4) (2, −3, −2)
(6, −9, −3) (3, −4, −2) (−3, 2, 3)

(−2, 0, −3)
x = 203 ;y = –25
16 16 ; z = –231
16 ;
(x, y, z)
x =5z +2;y =−3z +1;z
(x, y, z)
x =5z −2;y =4z −3;z
b ⎡ 3 −2 −1 −2
⎣ ⎢ ⎤
−2 1 0 5⎥
5 4 1 −1⎦
a ⎡ ⎣ 2 3 4 −2 −5 ⎤
2⎦
⎧2x −4y =−2
⎧2x −2y =−1
a ⎡ 2 −5 −3⎤
⎨
⎩3x −3y =−1
⎨2y − z =2
⎣4 −3 −1⎦
⎩
b ⎡ 3x − z =−2
4 3 −2 −3
⎣ ⎢ ⎤
−2 1 −3 4⎥
−1 −4 5 −2⎦
⎡1 −2 3 −4⎤
a ⎡ ⎣ 3 2 1 ⎤
4 −6 −3⎦
a ⎡ 2 3 ⎤
b ⎡ ⎣ 12 8 4 ⎤
⎣ ⎢6−5 ⎢0 5 −11 17⎥
3 −3 1 −1⎥
⎣0 1 −10 7⎦
2 1 −4 5 ⎦
4 −6 −3⎦
c ⎡ ⎣ 12 8 4 ⎤ b ⎡
24 −10 −5⎦
⎣ ⎢ 6 −5 2 3 ⎤
15 −15 5 −5⎥
2 1 −4 5 ⎦
c ⎡ 2 −7 10 −7
⎣ ⎢ ⎤
15 −15 5 −5⎥
2 1 −4 5 ⎦
(1, −1) (3, 3) (−2, 5, 2)
(−3, −5, 4) (−3, 2, 3) (−2, 0, −3)
(x, y, z)
(x, y, z)
x =5z +2;y =−3z +1;z
x = 1 2 z +4;y = 1 2 z −6;z
a b −14 c −6 a b −3 c
−77 −24
(7, 6) (−2, 0)
(−3, 2) (−9, 3) (−3, −5, 4)
(2, −3, −2) (−3, 2, 3) (−2, 0, −3)
a b

a b a b

⎧f ≥0
⎪p ≥0
a ⎨
⎪f + p ≤20
⎩2 f +5p ≤50
b
⎧c ≥0
⎪a ≥0
a ⎨
⎪c + a ≤24
⎩a ≥3c
b
⎧w ≥0
⎪
b ≥0
a ⎨
⎪27w +16b >80
⎩3.20w + 1.75b ≤10
b
c
d
c
d
c
d

⎧w ≥0
⎪r ≥0
a ⎨
⎪w + r ≥4
⎩270w + 650r ≥ 1500
b
c
d
a b m =3; b =−12 (6, 0), (0, − 4)
11
6,287.05 0.082 $5.85
7
x =1 p = 8.4 41
−6
−4
3
2
−4 < a
a b
(3, −1)
(4, 5) (3, 1)
(4, −1) (6, 2)
1
2

46 2 3
23 1
3
a b
(−3, 2, −4)
⎡4 3 0 −2⎤
⎧x −3z =−1
⎢1 −2 −3 7⎥
⎨x −2y =−2
⎣2 −1 2 −6⎦
⎩−y +2z =3
a ⎡ 1 −3 −2 4
⎣ ⎢ ⎤
4 −2 −3 −1⎥
2 2 −1 −3⎦
b ⎡ −6 −4 8⎤
⎢24 −2 −3 −1⎥
⎣2 2 −1 −3⎦
c ⎡ −6 −4 8⎤
⎢24 −2 −3 −1⎥
⎣0 −6 −1 5⎦
(−2, 5, −2) −4
(−3, 2) (−3, 2, 3)
a b
⎧b ≥0
⎪n ≥0
a ⎨
⎪b + n ≤40
⎩12b +18n ≥ 500
b
c
d

(2, 1) (2, −2, 1)
(2, −3)
(5, 7, 4) 7
⎧C ≥0
⎪L ≥0
a ⎨
⎪C + 0.5L ≤50
⎩L ≥3C
b
c
d
11x 2 +8x +6 3n +9 −200
−8
x
3y
−10.946
2x +6 a 81 b 81 c −81 21
3+x
d
6y 2 11a 2 +7a +5
a
b c
d e
a b
c d
e
a 21q 2 b 13mn
3

a −7c
2 b −10y 2 z 3 a 5y 2 +3z 2
a −4m 2 + n 2
b m 2 n 2 −8m 2 +4n 2 b pq 2 −6p −5q 2
8x 2 −11x +8 17y 2 +14y +1 x 2 +3x −5
6b 2 +3 −5ab +7b 2 7mn +4n 2
5x 2 −5xy +5y 2 7x 2 −6xy −2y 2 x 3 + y 3
p 3 −3p 2 q + q 3 a b c −15 a b c −4
150 a ( f + g)(x) =3x 2 −6x −3
a ( f + g)(x) =6x 2 − x +7
b ( f + g)(3) = 58
c ( f − g)(x) =4x 2 −7x −9
d ( f − g)(−2) = 21
a b
17 b 4
3x c 12p 6
d x
18
b ( f + g)(3) = 6
c ( f − g)(x) =x 2 −2x +9
d ( f − g)(−2) = 17
a x
16 b 10
x +1 a 12z 8
d b
19
a x
5 b 6
9 c 1 x 4
a y 6 b 10
8 c 1 m 8
a b
d 1
12 15 d 1
9 11
a b a 1 z 3 b 1
10 7 c p 8 d 64 a 1 n 2 b 1
10,000 c q7
d 16
a 81
16 b n2
m 2 a 125
27 b b4
a 4 a 1 z 9 b 1
p 3 q 3 c − 12v5 u
a 1
c 15 b 1
r 2 s 8 c 30d3
c 8 a b
35 b 5
12 c a
48 a z 54 b 3
49 c q 29
a 32w 5 x 5 b c 1
16b 12 a −27y 3 b c 1
d 64
a 8 d 8
16x 8 a p4
c 12
10000 b n7
m 7
c 81a4 b 12
c 8 d 27
x 6 y 9
a −8
q 3 b x4 c x2 y 6
w 4 9z 4 a 81c 24 d 30 b 1
a 18
a 256a 22 b 24 b 1
c
d 8n6
y 15 c 2x 3 y 10
m 6
a 9.6 × 10
4 b 7.8 × 10
−3 a 4.83 × 10
4
b 1.29 × 10
−2
p 39
a b −0.00012
a −950,000 b a −0.006 b a −0.009 b
a −35y 11 b 6a 5 b 6 a 54b
9 b 8r 11 s 8 a −15y 3 −24y 2 +21y
a 8x 4 −24x 3 +20x 2
b −18a 5 b +12a 4 b 2 −36a 3 b 3 a x 2 +17x +72
b 15c 2 +14c −8
b 12x 4 y 2 −20x 3 y 3 +12x 2 y 4
a 20x 2 +51x +27
b 30y 2 −3y −6

a x 2 −2x −35
a b 2 +3b −18
a x 3 −8x 2 +6x −48
b 15x 2 +29x −14
a y 3 −9y 2 +7y −63
b 16y 2 −20y −50
b 8a 2 b 2 +12ab −20
15m 2 −51m +42 42b 2 −53b +15
b 8x 2 y 2 +2xy −15
a y 3 −8y 2 +17y −6
b y 3 −8y 2 +17y −6
a y 2 +22y + 121
b 16x 2 −40xy +25y 2
a 18b 2 +77b −18
b
81p 2 −72p +16
c
49y 2 +14y +1
d
16r 2 −9
a
⎛
⎝f · g ⎞ ⎠(x) = x 3 + x 2 −52x −28
b ⎛ ⎝f · g ⎞ ⎠(2) = −120
4y 2
7x 4
a 2x 3 +5x 2 −7x +20
b y 3 −8y 2 +17y −6
a x 2 +18x +81
b 4c 2 −4cd + d 2
a 36x 2 −25
a 4x 2 −49 b 9x 2 − y 2
b 16p 2 −49q 2
a
36x 2 +84x +49 b
9x 2 −16 c
10x 2 −29x +10 d
36n 2 −12n +1
− 9
a 5 b
a
⎛
⎝f · g ⎞ ⎠(x) = x 3 −7x 2 +13x −15
b ⎛ ⎝f · g ⎞ ⎠(2) =−9
−9d
c 4
5
−4a +2b
8m 5 n 3
8a 5 b +6a 3 b 2 y +7 m +5
x 3 −3x 2 +2x +1+ 3
x +3 x 3 −3x 2 −2x −1− 3
x +3
x 2 +4x +16
25x 2 +10x +4 3x 2 +4x −2;2 4x 2 −3x +1;1
x 3 −4x 2 +5;0 x 3 −3x 2 +2;0
a ⎛ ⎝g f ⎞ ⎠
(x) = x −9
−1 6
b ⎛ ⎝g f ⎞ ⎠ (−5) = −14
a ⎛ ⎝g f ⎞ ⎠
(x) = x −8
b ⎛ ⎝g f ⎞ ⎠
(−3) = −11
a b
c d
e
a binomial,1 b trinomial,2 c
polynomial,3 d binomial,3
e monomial,0
a b
c d
e
a 12x
2 b −5a
a trinomial,2 b monomial,4
c binomial,1 d polynomial,3
e monomial,0
a 6w b 19x 2 y

12x 2 −5a 6w +19x 2 y a −22b b 16xy
a −10a +5b
a −4a 2 + b 2
a xy 2 −5x −5y 2
b pq 2 −4p −3q 2 b x 2 y −3x +7xy 2 b 19y +5z
x −3y + a 2 b −4a −5ab 2 −u 2 +4v 2 + 12a+8b −4a − 3b
−17x 6 11y 2 +4y +11 −3x 2 +17x −1
11x 2 −5x +5 6a 2 −4a −1 2m 2 −7m +4
11a +3 12s 2 −16s +9 3x 2 − x +4
w 2 +3w +4 11w −66 10x 2 −7xy +6y 2
10m 2 +3mn −8n 2 −3ab +3b 2 p 3 −6p 2 q + pq 2 +4q 3
x 3 +2x 2 y −5xy 2 + y 3 a b c a −104 b c
a ( f + g)(x) =7x 2 +4x +4
a ( f + g)(x) =6x 3 − x 2 −9x +3
b ( f + g)(2) = 40
c ( f − g)(x) =−3x 2 −12x −2
d ( f − g)(−3) = 7
b ( f + g)(2) = 29
c ( f − g)(x) =−x 2 +5x +3
d ( f − g)(−3) = −21
a d
9 b 4
14x c 8y 4 d w
6 a n
31 b 3
x +6 c 56w
6
d a
16
m x +3 y a + b a x
15 b 5
9 c 1
q 18 d 10
1
a p 14 b 4
12 c 1 b 8 d 1 4 5 a b a −1 b −1
a 1 a 2 b 1
1000 c c 5 d 9 a 1 r 3 b
1
100,000 c q10
a 729
64 b − v5
u 5
a 1 b 4 b w2
x 9 c −12cd
4
d 1,000
a 1 25 b − 1 25 c 25 d
−25
a b 1
u 4 v 5 c −36r2
s 5
a 64
25 b a2
b 2
a 3 5 b 1 15
a m
8 b 10
18 c 1
x 12 a y 3x b 5
xy
c 1
q 48 a 9x 2 y 2 b c 1
25x 4
a −125a 3 b 3 b c 1
36x 6
d 9 y 8
a p5 b y6 c 8x3 y 6
32 x 6 z 3
d 16
p 6 q 4
1 p
d 16
y 6
a a4 b 81b 4 16m2
25
d q8 r 8
p 2
a 1125t
8 b 1
t 19 c y4
3x 2 a 16m 8 n 22 b 4 p 6 a 7 n b 1 7n c − 1 7n
a 1
9p 2 b 3 p 2 c −3
p 2 x 14 a 36
8m 18 1,000x 6 y 3 16a 12 b 8
c a4 c 4
9b 6

8
27 x6 y 3 1,024a 10 25,000p 24
x 18 y 18 144m 8 n 22 a 45x
3 b 48y 4
a 1
2r 4 b 1 3 x11
a 3.4 × 10
5 b 4.1 × 10
−2 a 1.29 × 10
6
a
b
1
j 3
b 1.03 × 10
−6
a b
−4000
n 12
a −830 b
a b
a 30x
8 b 15x 8 y 2
a 72c
5 b 1 3 m5 n 5 a −5t 3 −15t 2 +90t
b 9sr 5 −27s 2 r 4 +45s 3 r 3 a −5m 3 −15m 2 +90m
b −21x 4 y 2 −30x 3 y 3 +3x 2 y 4
y 2 +12y +27 21q 2 −44q −32 y 2 −8y +12
6p 2 +11p +5 m 2 +7m −44 33r 2 −85r −8
y 3 +3y 2 −4y −12 6x 2 y 2 +13xy +6 y 4 −11y 2 +28
9r 2 s 2 −33rs +28 u 3 +7u 2 +14u +8 3a 3 +31a 2 +5a −50
6a 4 −13a 3 +15a 2 +35a −50
p 3 −10p 2 +33p −36
6r 3 −41r 2 −61r −9
q 2 +24q + 144 4y 2 −12yz +9z 2 x 2 + 4 3 x + 4 9
1
64 x2 −
36 1 xy + 81 1 y2 25u 4 +90u 2 +81 64p 6 −48p 3 +9
64j 2 −16 81c 2 −25 49w 2 − 100x 2
p 2 − 16
25 q2 x 2 y 2 −81 225m 4 −64n 8
t 2 −18t +81 2x 2 −3xy −2y 2 9p 2 −64
k 2 −12k +36 8x 3 − x 2 y 2 +64xy −8y 3 y 8 +4y 4 z +4z 2
m 6 −16m 3 n +64n 2 r 5 + r 2 s 2 − r 3 s 3 − s 5 18p −9
−10j −15 9d 7 42m 5 n 8
5q 5 −10q 4 +30q 3 y 3 − y 2 −2y 6k 3 +11k 2 −26k +4
121 − b 2 4x 4 −9y 8 9d 2 +6d +1
9z 2 + 6 5 z + 1 25
a
⎛
⎝f · g ⎞ ⎠(x) = x 4 +6x 3 +9x 2 +10x −12
b ⎛ ⎝f · g ⎞ ⎠(1) =14
2m 3
3n 5
a ⎛ ⎝f · g ⎞ ⎠(x) =4x 3 − x 2 −8x +5
b ⎛ ⎝f · g ⎞ ⎠(−2) = −15
−3y 3
4x 3
a ⎛ ⎝f · g ⎞ ⎠(x) =49x 2 −64
b ⎛ ⎝f · g ⎞ ⎠(−2) = 132
−3q 5
5v 4
4x 2 +3x −6y 2 +3y
u 2
p 5

6r 3 +11r 2 s −8rs 3
4x +3
−5y −3+ 1 4y
a −7
p +3− 8
p +8 2n 2 −6n +8+ 4
n +3
m 2 −10m + 100 25y 2 +20y +16 x 2 −5x +6;0
2x 2 −3x +4;4 x 3 −2x 2 +5x −4;−2 3x 3 −2x 2 −4x −2;0
a ⎛ ⎝g f ⎞ ⎠
(x) = x −6
b ⎛ ⎝g f ⎞ ⎠
(−5) = −11
a ⎛ ⎝ f g ⎞ ⎠ (x) = x2 +5x −4
b ⎛ ⎝g f ⎞ ⎠
(0) =−4
a
⎛
⎝f · g ⎞ ⎠(x) = x 4 +6x 3 +9x 2 +10x − 12;
b ⎛ ⎝f · g ⎞ ⎠(1) =14
−9
−13y 3
6m 2 +19m −4 5u 2 +4u +7 14x −7
−7b −3a 8a 2 −5a −2 3m 2 −3m +3
3y 2 −8y −14 x 3 +2x 2 y −5xy 2 + y 3 a b c
64 feet
a ( f + g)(x) =4x 2 −5x −2
b ( f + g)(3) = 19
c ( f − g)(x) =−3x −12
d ( f − g)(−2) = −6
p 13
a 6 y a + b a 5
1
x 5 −3
−
1000 1
1
8n
1
1000
−125
−
125
1
−x 3
27
3 10 x 6
n 2
1
u 3 v 5
1 p
81x 4 y 8
27x 7 y 17 3y 4
z 4
4x 4
5.3 × 10 6 29,000 0.00009413
0.00072 9,000 10c 10
m 7 n 10
9
1
q 16
9
y 8
a 4 −9a 3 −36a 2 8n 4 −10n 3
y 2 +8y −48 18p 2 −93p + 110 k 2 −3k −54
10y 2 −59y +63 x 2 + x −72 30a 2 −33a +3
15b 3 − b 2 −47b +18 24y 3 −54y 2 +32y −5 x 2 + 3 2 x + 16
9
25p 2 +70pq +49q 2 36x 2 − y 2 144x 6 −49y 4

9p 9 − 3 y 4 4a 5
b 2
4m 9
n 4 −9y 2 +4y −3x −1+
4x
3
y −3+ 33
y +5 a 2 − a +1− 2
a +1
a ⎛ ⎝g f ⎞ ⎠
(x) = x −6
−9
b ⎛ ⎝g f ⎞ ⎠ (−2) =−8
a b 6x 2 −3x +11 x
1
1
8y 3 x 9
2x 2 −5x −4;0
4r
s 6 −48x 5 y 9 7m 2 +19m −6
16x 2 −24x +9 3y 2 −7x
4.8 × 10 3 a 36 b 21 c −9 a ⎛ ⎝g f ⎞ ⎠
(x) =3x +4
b ⎛ ⎝g f ⎞ ⎠
(3) =13
2·2·2·7 −21a 2 −24ab
6y 2 +23y +20
27x 6 m 2 +8m +16 x 2 −9
y = 3 5
n(n − 11)(n +2) 8; x =2
5m 2
7x
3y 2 ⎛ ⎝3x +2x 2 +7y ⎞ ⎠
3p ⎛ ⎝p 2 −2pq +3q 2⎞ ⎠ 2x 2 (x +6) 3y 2 ⎛ ⎝2y −5 ⎞ ⎠
3xy ⎛ ⎝5x 2 − xy +2y 2⎞ ⎠ 2ab ⎛ ⎝4a 2 + ab −3b 2⎞ ⎠ −4b ⎛ ⎝b 2 −4b +2 ⎞ ⎠
−7a ⎛ ⎝a 2 −3a +2 ⎞ ⎠ (m +3)(4m −7) (n −4)(8n +5)
(x +8) ⎛ ⎝y +3 ⎞ ⎠ (a +7)(b +8) a (x −5)(x +2)
a ⎛ ⎝y +4 ⎞ ⎠ ⎛ ⎝y −7 ⎞ ⎠
b (7m −3)(6m −5)
b (5x −4)(4x −3)
⎛ ⎝q +4 ⎞ ⎠ ⎛ ⎝q +6 ⎞ ⎠ (t +2)(t +12)
(u −3)(u −6) ⎛ ⎝y −7 ⎞ ⎠ ⎛ ⎝y −9 ⎞ ⎠ (m +3)(m +6)
(n −3)(n −4) (a − b)(a −10b) (m − n)(m −12n)
5x(x −1)(x +4)
6y ⎛ ⎝y −2 ⎞ ⎠ ⎛ ⎝y +5 ⎞ ⎠ (a +1)(2a +3) (b +1)(4b +1)

(2x −3)(4x −1) ⎛ ⎝2y −7 ⎞ ⎠ ⎛ ⎝5y −1 ⎞ ⎠ ⎛ ⎝3x +2y ⎞ ⎠ ⎛ ⎝6x −5y ⎞ ⎠
⎛ ⎝3x + y ⎞ ⎠ ⎛ ⎝10x −21y ⎞ ⎠ 5n(n −4)(3n −5) 8q ⎛ ⎝q +6 ⎞ ⎠ ⎛ ⎝7q −2 ⎞ ⎠
(x +2)(6x +1) ⎛ ⎝2y +1 ⎞ ⎠ ⎛ ⎝2y +3 ⎞ ⎠ 4(2x −3)(2x −1)
3(3w −2)(2w −3)
⎛ ⎝h 2 −2 ⎞ ⎠ ⎛ ⎝h 2 +6 ⎞ ⎠
⎛ ⎝y 2 +4 ⎞ ⎠ ⎛ ⎝y 2 −5 ⎞ ⎠
(x −3)(x −1) ⎛ ⎝y −1 ⎞ ⎠ ⎛ ⎝y +1 ⎞ ⎠
(2x +3) 2
⎛ ⎝3y +4 ⎞ 2
⎠
⎛ ⎝7x +6y ⎞ 2
⎠
⎛ ⎝8y −5 ⎞ 2
⎠
(4z −9) 2
(8m +7n) 2 2y(2x −3) 2
3q ⎛ ⎝3p +5 ⎞ ⎠2 (11m −1)(11m +1) ⎛ ⎝9y −1 ⎞ ⎠ ⎛ ⎝9y +1 ⎞ ⎠
(14m −5n)(14m +5n) ⎛ ⎝11p −3q ⎞ ⎠ ⎛ ⎝11p +3q ⎞ ⎠
2y 2 (x −2)(x +2) ⎛ ⎝x 2 +4 ⎞ ⎠
7c 2 (a − b)(a + b) ⎛ ⎝a 2 + b 2⎞ ⎠ ⎛ ⎝x −5−y ⎞ ⎠ ⎛ ⎝x −5+y ⎞ ⎠ ⎛ ⎝x +3−2y ⎞ ⎠ ⎛ ⎝x +3+2y ⎞ ⎠
(x +3) ⎛ ⎝x 2 −3x +9 ⎞ ⎠
125(2m − n)(4m 2 +2mn + n 2 )
⎛ ⎝y +2 ⎞ ⎛
⎠⎝y 2 −2y +4 ⎞ ⎠
4(5p + q) ⎛ ⎝25p 2 −5pq + q 2⎞ ⎠
⎛ ⎝2x −3y ⎞ ⎛
⎠⎝4x 2 +6xy +9y 2⎞ ⎠
2(6c +7d) ⎛ ⎝36c 2 −42cd +49d 2⎞ ⎠
⎛ ⎝−2y +1 ⎞ ⎛
⎠⎝13y 2 +5y +1 ⎞ ⎠ (−4n +3) ⎛ ⎝31n 2 +21n +9 ⎞ ⎠ 8y ⎛ ⎝y −1 ⎞ ⎠ ⎛ ⎝y +3 ⎞ ⎠
5y ⎛ ⎝y −9 ⎞ ⎠ ⎛ ⎝y +6 ⎞ ⎠ 4x(2x −3)(2x +3) 3 ⎛ ⎝3y −4 ⎞ ⎠ ⎛ ⎝3y +4 ⎞ ⎠
⎛ ⎝2x +5y ⎞ 2
⎠
3xy ⎛ ⎝9y 2 +16 ⎞ ⎠
⎛ ⎝3x −4y ⎞ 2
⎠
2(5m +6n) ⎛ ⎝25m 2 −30mn +36n 2⎞ ⎠
2xy ⎛ ⎝25x 2 +36 ⎞ ⎠
2 ⎛ ⎝p +3q ⎞ ⎛
⎠⎝p 2 −3pq +9q 2⎞ ⎠
4ab ⎛ ⎝a 2 +4 ⎞ ⎠(a −2)(a +2) 7xy ⎛ ⎝y 2 +1 ⎞ ⎠ ⎛ ⎝y −1 ⎞ ⎠ ⎛ ⎝y +1 ⎞ ⎠ 6(x + b)(x −2c)
2(4x − 1)(2x +3y) 4q ⎛ ⎝p −3 ⎞ ⎠ ⎛ ⎝p −1 ⎞ ⎠ 3p ⎛ ⎝2q +1 ⎞ ⎠ ⎛ ⎝q −2 ⎞ ⎠
⎛ ⎝2x −3y −5 ⎞ ⎠ ⎛ ⎝2x −3y +5 ⎞ ⎠
⎛ ⎝4x −3y −8 ⎞ ⎠ ⎛ ⎝4x −3y +8 ⎞ ⎠
m = 2 3 , m =−1 2
p =− 3 4 , p = 3 4 c =2,c = 4 3 d =3,d =− 1 2
p = 7 5 ,p= −7 5 x = 11 6 , x =−11 6 m =1,m = 3 2
k =3,k =−3
x =0,x = 3 2 y =0,y = 1 4
a x =1 x =7
b (1, −4) (7, −4)
a =− 5 2 , a = 2 3 b =−2,b =− 1 20
a x =1 x = 5 2
⎛
b (1, 0), ⎝ 5 2 ,0⎞ ⎠ c (0, 5)
a x =−3 x =5
b (−3, 7) (5, 7)
a x =−3 x = 5 6
⎛
b (−3, 0), ⎝ 5 6 ,0⎞ ⎠ c (0, −15)
−15, −17 −23, −21
a b
a b
c
c
2pq 6m 2 n 3 2a
5x 3 y 3(2m +3) 9(n −7)

3(x 2 +2x −3) 2(4p 2 +2p +1) 8y 2 (y +2)
5x(x 2 −3x +4) 3x ⎛ ⎝8x 2 −4x +5 ⎞ ⎠ 6y 2 (2x +3x 2 −5y)
4xy ⎛ ⎝5x 2 − xy +3y 2⎞ ⎠ −2(x +2)
−2x ⎛ ⎝x 2 −9x +4 ⎞ ⎠
−4pq ⎛ ⎝p 2 +3pq −4q ⎞ ⎠ (x +1)(5x +3) (b −2)(3b −13)
(b +5)(a +3) ⎛ ⎝y +5 ⎞ ⎠ ⎛ ⎝8y +1 ⎞ ⎠ (u +2)(v −9)
(u −1)(u +6) ⎛ ⎝3p −5 ⎞ ⎠ ⎛ ⎝3p +4 ⎞ ⎠ (n −6)(m −4)
(x −7)(2x −5) −9xy ⎛ ⎝2y +3x ⎞ ⎠
(x 2 + 2)(3x −7)
(x + y)(x +5)
⎛ ⎝p +5 ⎞ ⎠ ⎛ ⎝p +6 ⎞ ⎠ (n +3)(n +16) (a +5)(a +20)
(x −2)(x −6) (y −3)(y − 15) (x −1)(x −7)
(p −1)(p +6) (x −4)(x −2) (x − 12)(x +1)
(x +8y)(x −10y) (m + n)(m −65n) (a +8b)(a −3b)
p(p − 10)(p +2)
3m(m −5)(m −2) 5x 2 (x −3)(x +5) (2t +5)(t +1)
(11x +1)(x +3) (4w −1)(w −1) ⎛ ⎝4q +1 ⎞ ⎠ ⎛ ⎝q −2 ⎞ ⎠
(2p −5q)(3p −2q) (4a −3b)(a +5b) −16(x +1)(x +1)
−10q ⎛ ⎝3q +2 ⎞ ⎠ ⎛ ⎝q +4 ⎞ ⎠ (5n +1)(n +4) (2k − 3)(2k −5)
(3y + 5)(2y −3) (2n +3)(n − 15) 10(6y −1)(y +5)
3z(8z + 3)(2z −5) 8(2s +3)(s +1) 12 ⎛ ⎝4y −3 ⎞ ⎠ ⎛ ⎝y +1 ⎞ ⎠
⎛ ⎝x 2 +1 ⎞ ⎠ ⎛ ⎝x 2 −7 ⎞ ⎠ (x 2 −7)(x 2 +4) (x − 12)(x +1)
⎛ ⎝3y −4 ⎞ ⎠ ⎛ ⎝3y −1 ⎞ ⎠ (u −6)(u −6) (r −4s)(r −16s)
⎛ ⎝4y −7 ⎞ ⎠ ⎛ ⎝3y −2 ⎞ ⎠ (2n −1)(3n +4)
13 ⎛ ⎝z 2 +3z −2 ⎞ ⎠
3p ⎛ ⎝p +7 ⎞ ⎠ 6(r +2)(r +3) 4(2n +1)(3n +1)
⎛ ⎝x 2 +2 ⎞ ⎠ ⎛ ⎝x 2 −6 ⎞ ⎠ (x −9)(x +6)
(4y +3) 2 (6s +7) 2
(10x −1) 2 (5n − 12) 2 ⎛ ⎝7x +2y ⎞ 2
⎠
⎛ ⎝10y −1 ⎞ 2
⎠
10j(k +4) 2 3u 2 (5u − v) 2
(5v −1)(5v +1) (2 − 7x)(2 + 7x) 6p 2 (q −3)(q +3)
6 ⎛ ⎝4p 2 +9 ⎞ ⎠ ⎛ ⎝11x −12y ⎞ ⎠ ⎛ ⎝11x +12y ⎞ ⎠ (13c −6d)(13c +6d)
(2z − 1)(2z + 1)(4z 2 +1)
2b 2 (3a −2)(3a +2) ⎛ ⎝9a 2 +4 ⎞ ⎠
⎛ ⎝x −8−y ⎞ ⎠ ⎛ ⎝x −8+y ⎞ ⎠
(a +3−3b)(a +3+3b) (x +5)(x 2 −5x + 25) (z 2 −3)(z 4 +3z 2 +9)
(2 − 7t)(4 + 14t +49t 2 )
(2y −5z)(4y 2 +10yz +25z 2 )
(6a +5b)(36a 2 −30ab +25b 2 )
7(k +2)(k 2 −2k +4) 2x 2 (1 − 2y)(1 + 2y +4y 2 ) 9(x +1) ⎛ ⎝x 2 +3 ⎞ ⎠
− ⎛ ⎝3y +5 ⎞ ⎛
⎠⎝21y 2 −30y +25 ⎞ ⎠ (8a −5)(8a +5) 3 ⎛ ⎝3q −1 ⎞ ⎠ ⎛ ⎝3q +1 ⎞ ⎠
(4x −9) 2 2 ⎛ ⎝4p 2 +1 ⎞ ⎠ ⎛ ⎝5−2y ⎞ ⎛
⎠⎝25 + 10y +4y 2⎞ ⎠
5(3n +2) 2 ⎛ ⎝x −5−y ⎞ ⎠ ⎛ ⎝x −5+y ⎞ ⎠
(3x +1) ⎛ ⎝3x 2 +1 ⎞ ⎠
(2n −1)(n +7)
a 3 (a 2 +9) (11r − s)(11r + s) 8(m −2)(m +2)

(5w −6) 2 (m +7n) 2 7(b +3)(b −2)
3xy(x −3) ⎛ ⎝x 2 +3x +9 ⎞ ⎠ (k −2)(k +2)(k 2 +4) 5xy 2 ⎛ ⎝x 2 +4 ⎞ ⎠(x +2)(x −2)
3(5p +4)(q −1) 4(x +3)(x +7)
4u 2 (u + v) ⎛ ⎝u 2 − uv + v 2⎞ ⎠
10(m −5)(m +5) ⎛ ⎝m 2 +25 ⎞ ⎠ 3y(3x +2)(4x −1)
⎛ ⎝2x −3y ⎞ ⎛
⎠⎝4x 2 +6xy +9y 2⎞ ⎠
⎛
⎝y +1 ⎞ ⎠ ⎛ ⎝y −1 ⎞ ⎛
⎠⎝y 2 − y +1 ⎞ ⎠ ⎛ ⎝y 2 + y +1 ⎞ ⎠
⎛ ⎝3x − y +7 ⎞ ⎠ ⎛ ⎝3x − y −7 ⎞ ⎠
(3x −2) 2
a =10/3,a =7/2 m =0,m =5/12 x =1/2
a = −4
5 , a =6 m =5/4,m =3 a =−1,a =0
m =12/7,m = −12/7 y =−9/4,y =9/4 n = −6/11, n = 6/11
x =2,x =−5 x =3/2,x =−1 x =2,x =−4/3
x =3/2 x =2,x =−4/3 x =−3/2,x =1/3
p =0, p =¾ x =0,x =6 x =0,x =–1/3
a x =2 x =6b (2, −4)
(6, −4)
a x = 5 3 x =− 1 2
b ⎛ ⎝ 5 3 ,0⎞ ⎠ ⎛
⎝ − 1 2 ,0⎞ ⎠ c (0, −5)
a x = 3 2 x = 3 4
b ⎛ ⎝ 3 2 ,−4⎞ ⎠ ⎛ ⎝ 3 4 ,−4⎞ ⎠
−13, −11
a x = 2 3 x =− 2 3
b ⎛ ⎝ 2 3 ,0⎞ ⎠ ⎛
⎝ − 2 3 ,0⎞ ⎠ c (0, −4)
−14, −12
a
b
3ab 2 3y 7(5y + 12)
3x(6x 2 −5) 4x(x 2 −3x +4) −3x ⎛ ⎝x 2 −9x +4 ⎞ ⎠
(a + b)(x − y) (x −3)(x +7) (m 2 +1)(m +1)
(a +3)(a +11) (m +9)(m −6) ⎛ ⎝x +5y ⎞ ⎠ ⎛ ⎝x +7y ⎞ ⎠
(a +7b)(a −3b) 3y(y −5)(y −2)
(5y +9)(y +1) ⎛ ⎝5y +1 ⎞ ⎠ ⎛ ⎝2y −11 ⎞ ⎠ −9(9a +1)(a −2)
(3a −1)(6a −1) 3(x +4)(x −3)
3(2a −7)(3a +1) (x 2 − 15)(x 2 +2) (5x +3) 2
10(2x +9) 2 3u 2 (5u − v) 2 (13m + n)(13m − n)
⎛ ⎝3+11y ⎞ ⎠ ⎛ ⎝3−11y ⎞ ⎠ n(13n +1)(13n −1)
6 ⎛ ⎝4p 2 +9 ⎞ ⎠
(2z − 1)(2z + 1)(4z 2 +1) (a +3−3b)(a +3+3b)
(a −5) ⎛ ⎝a 2 +5a +25 ⎞ ⎠
2(m +3) ⎛ ⎝m 2 −3m +9 ⎞ ⎠ 4x 2 (6x +11) (4n −7m) 2
5u 2 (u +3)(u −3) (b −4) ⎛ ⎝b 2 +4b +16 ⎞ ⎠
(2b +5c)(b − c) 5 ⎛ ⎝q +3 ⎞ ⎠ ⎛ ⎝q −6 ⎞ ⎠
10(m −5)(m +5) ⎛ ⎝m 2 +25 ⎞ ⎠
⎛ ⎝4x −3y +8 ⎞ ⎠ ⎛ ⎝4x −3y −8 ⎞ ⎠ b =−1/5,b =−1/6 x =1/2

x =−4,x =−5
p =− 5 2 , p =8 m = 5 12 , m =−5 12
x =2,x =−5 p =0, p =¾ a x =−7 x =−4
b (−7, −8) (−4, −8)
a x = 7 8 x =− 7 8
b ⎛ ⎝ 7 8 ,0⎞ ⎠ , ⎛
⎝ − 7 8 ,0⎞ ⎠ c (0, −49)
−21
−19
40a 2 (2 + 3a) (x +4)(x +9) (x −8) ⎛ ⎝y +7 ⎞ ⎠
(3s −2) 2 3 ⎛ ⎝x +5y ⎞ ⎠ ⎛ ⎝x −5y ⎞ ⎠
(x +5) ⎛ ⎝x 2 −5x +25 ⎞ ⎠
⎛ ⎝3x 2 −5 ⎞ ⎠ ⎛ ⎝2x 2 −3 ⎞ ⎠ a =4/5,a =−6
a x =3 x =4b (3, −7)
(4, −7)
6 y
4
25
15
4
37
30 27x − 32
36
8x +26
2
3
1
54 x =− 1 4
x =−1 n =9, n =−4
y = 10 − 5x
2
n =90
x = 10 7
a b c x >−2 [−3, 5)
a x =0b n =− 1 3
a q =0b y =− 2 3
c a =−1,a =−3
c m =2,m =−3
x −5
x −1 , x ≠−2, x ≠1 2 ⎛ ⎝x −3y ⎞ ⎠
3 ⎛ ⎝x +3y ⎞ ⎠
− x x +1
+5 − x x +2
+1
3(x −6)
x −4
x +5
2 ⎛ ⎝x 2 +2x +4 ⎞ 2z
⎠
(x +2) ⎛ ⎝x 2 z −1
−2x +4 ⎞ ⎠
2
y +5
R(x)
x ≠5
x ≠−1.
x −5 −
x +1, x ≠2, x ≠1
x −1
5(x − y)
2 ⎛ ⎝x +5y ⎞ ⎠
x −2
2(x +3)
(b +2)(b −1)
(1 + b)(b +4)
x +2
4
2(m +1)(m +2)
(n +5)(n +9)
3(m +4)(m −3) 2(n + 6)(2n +3)
R(x) R(x) =2
x ≠4
x ≠−2.

R(x) = 1 3 R(x) =
4(x x −2
x(x −2)
−8) R(x) =
x −1
x +2 x +3
x −11
x −2
x −3
x +9
a (x −4)(x +3)(x +4)
2x +8
b
(x −4)(x +3)(x +4)
x +3
(x −4)(x +3)(x +4)
7m +25
(m +3)(m +4)
1
x −2
y +3
y +4
v +3
v +1
y +3
y +2
a (x +2)(x −5)(x +1)
b
3x 2 +3x
(x +2)(x −5)(x +1)
5x −25
(x +2)(x −5)(x +1)
x 2 −3x +18
(x +3)(x −3)
14
11
ab
b − a
7
3
y − x
xy
3
5x +22
x(x +1)
3(x −1)
5m 2 −9m +2
(m +1)(m −2)(m +2)
−3
z −3
1
(b +1)(b −1)
3w
w +7
2
3(x −1)
10
23
b(b +2)(b −5)
3b −5
10
3
3(x −2)
5x +7
2 ⎛ ⎝2y 2 +13y +5 ⎞ ⎠
3y
x =−3,x =5 y =−2,y =6
3n −2
n −1
7x −4
(x −2)(x +3)
2n 2 +12n −30
(n +2)(n −5)(n +3)
5x +1
(x −6)(x +1)
1
(x +2)(x +1)
x −7
x −4
1
2(x −3)
y + x
y − x
3
c +3
b + a
a 2 + b 2
x +21
6x −43
x
x +4
y =− 15 7 x = 15
13
x = 2 3
y =2
x =3 y =7
a
x ≠3
a
x ≠1
y = mx −4m +5
x ≠4.b x =2,x = 14 3
x ≠5.b x = 21 4 c ⎛ ⎝ 21 4 ,4⎞ ⎠
c (2, 3), ⎛ ⎝ 14 3 ,3⎞ ⎠
y = mx +5m +1
a =
b
cb −1 y =
x 3x
+6
y =33 z =14

a c = 4.8t b
a d =50t b
a h = 130
t
b 1 2 3
a x = 3500 p b (−∞, −4) ∪ [2, ∞) (−∞, −2] ∪ (4, ∞)
⎛ ⎝ −3 2 ,3⎞ ⎠
(−8, 4) (−∞, −4) ∪ (2, ∞)
(−∞, −4) ∪ (3, ∞) (2, 4) (3, 6)
(−4, 2] [−1, 4)
a c(x) = 20x + 6000
x
b
a c(x) = 5x + 900
x b
a z =0b p = 5 6
c n =−4,n =2
2m 2
3n
a +2
a +8
3(m +5n)
4(m −5n)
w 2 −6w +36
w −6
x 3
8y
a y =0 b x =− 1 2 c
u =−4,u =7
− 4 5
8
3
−1
p 2 +4
p −2
− z −5
4+z
4(b +9)
−
3(b +7)
− 1
v +5
x −2
8x(x +5)
4(m +8)(m +7)
3(m −4)(m +2)
p(p −4)
2 ⎛ ⎝p −9 ⎞ ⎠
x +5
x −1
4b(b −4)
(b +5)(b −8)
− 5
y +4
3
10
y −5
3(y +5)
(3c −1)(c +5)
(m −2)(m −3)
(3c +1)(c −5) −
(3 + m)(m +4)
3s
s +4
2a −7
5
(4p +1)(p −4)
3p(p +9)(p −1)
x ≠2 x ≠−3 R(x) =2
4(p 2 − pq + q 2 )
(p − q)(p 2 + pq + q 2 )
3(3c −5)
x ≠5 x ≠−5
R(x) = x +5
2x(x +2)

3x(x +7)
R(x) =
x −7
r +8
m −2
2
4 x +2
4b −3
b −7
a (b +3)(b +3)(b −5)
b
4b −20
(b +3)(b +3)(b −5)
2b 2 +6b
(b +3)(b +3)(b −5)
x(x −5)
R(x) =
x −6
3
5
3c +5
4c −5
2w
w −4
3a +7
p +3
r +9
p +5
r +7
a (x +2)(x −4)(x +3)
b
5x +15
(x +2)(x −4)(x +3)
2x 2 +4x
(x +2)(x −4)(x +3)
a (d +5)(3d −1)(d −6)
b
5r −7
(r +4)(r −5)
b(5b +10+2a 2 )
a 2 (b −2)(b +2)
2(7t −6)
(t −6)(t +6)
p +2
p +3
2d −12
(d +5)(3d −1)(d −6)
5d 2 +25d
(d +5)(3d −1)(d −6)
11w +1
(3w −2)(w +1)
− m m
+4
x −5
(x −4)(x +1)(x −1)
c −5
c +2
3(3x +8)
a
(x −8)(x +8)
b R(x) =
x +8
3
a −4
2a
4a 2 +25a −6
(a +3)(a +6)
3
r −2
z +4
z −5
a (z −2)(z +4)(z +2)
9z +18
b
(z −2)(z +4)(z +2)
4z 2 +16z
(z −2)(z +4)(z +2)
21y +8x
30x 2 y 2
1
(x −1)(x +1)
2y 2 + y +9
(y +3)(y −1)
3(r 2 +6r + 18)
(r +1)(r +6)(r +3)
−6
m −6
4(8x +1)
10x −1
5a 2 +7a −36
a(a −2)
3(d +1)
(x +8)(x +1)
d +2 a R(x) =−
(x −2)(x +3)
b R(x) = x x +1
+3
20
57
(x +1)(x −3)
2
19
2x −10
3x +16
2c +29
5c
1
2(c −2)
n 2 + m
m − n 2
4
a +1
c 2 + c
c − d 2
3z −19
3z +8
2p −5
5
a
b
c
d x + y
xy
12
13
rt
t − r
11
8
pq
q − p
4
3a −7
m(m −5)
(4m −19)(m +5)

13
24
2(a −4)
3mn
n − m
(x −1)(x −2)
6
a =10
v = 40
21
m =−2,m =4 p =−5, p =−4
v =14
x =− 4 5
z = −145
q = −18, q =−1 no solution no solution
b =−8 d =2 n =1
no solution
s = 5 4 x =− 4 3
a
x ≠−2
x ≠−4.b x =−3,x =− 14 5 c
(−3, 5), ⎛ ⎝ −14 5 ,5⎞ ⎠
a
x ≠2
x ≠5.b x = 9 2 , c ⎛9 ⎝2 ,2⎞ ⎠
r =
2π
C w =2v +7
c = b +3+2a a
q
p =
4q −2 w =
10 15v
+ v n = 5m +23
4
c = E m 2 y =
12 20x
− x
x =49 p = −11 a =16
m =60 p =30 a b
9 159 325
3 a b 12
2
3
8
247.3 160 29
30 20 4
60 650 50
50 2
2 44 7 30 10
y = 14 3 x p = 3.2q a P = 2.5g b $82.50
a m =8v b 16 a L =3d
2 b 300 y = 20 x
v = w 3 a g = 92,400
w
b a t = 1000 r
a c = 2 t
b 1 a c = 2.5m b
(−∞, −4) ∪ [3, ∞) [−1, 3)
(−∞, 1) ∪ (7, ∞) (−5, 6)
⎛ ⎝ −5 2 ,5⎞ ⎠
b 2.5
(−∞, −3) ∪ (6, ∞) [−9, 6) (−∞, −6] ∪ (4, ∞)
(−∞, −4) ∪ (−3, ∞) (1, 4)
(−∞, −3) ∪ ( 5 2 ,∞)
(−∞, 2 3 )∪(3 2 ,∞) (−∞, 0) ∪ (0, 4) ∪ (6, ∞) [−2, 0) ∪ (0, 4]
(−4, 4) [−10, −1) ∪ (2, ∞) (2, 5]
(−2, 6]

a ≠ 2 3
x +3
x +4
y ≠0
1
6
3x
(x +6)(x +6) −
11 1
− w
3
4
−3x
2
5
c +4
R(x) =3 1 y +5
x +4
q 2 +9
15w +2
(q +5)(q +1)
6w −1
3b 2 +19b −16
b 2 −49
11c −12
5x 2 +26x
(c −2)(c +3)
(x +4)(x +4)(x +6)
2m −7
m +2
R(x) = 2
x +11
(a +2)(a −5)(a +4) (3p −1)(p +6)(p +8)
11
8
b = 3 2
2(y 2 +10y −2)
(y +2)(y +8)
4
a −8 R(x) = x x +8
+5
x −2
(x +2)(x −5)
2x
2
z −5
21z +21
x = 6 7
a
x ≠2
x ≠4.b x =1,x =6
c (1, 1), (6, 1)
l =
hw
V z = y +5+7x
x x = 12 5
s =15
1126
b =9;x =2 1 3
45 16
48 12 x =7
301 288 9
(−4, 3] [−6, 4) (−∞, −2] ∪ [4, ∞)
(−∞, 2) ∪ [5, ∞)
a
3b
3n −2
n −1
a c(x) = 150x + 100000
x
b
$160
x +3
x −3
3x
x +9
n − m
m + n z = 1 2
[−3, 6) (−∞, 0) ∪ (0, 4] ∪ [6, ∞)
R(x) = 1
(x +2)(x +2)
⎛ ⎝ −3, 5
2
⎞
⎠ y = 81
16 4 4 5

a b c a x 6 b y 6 c z 12
x 5
53
60
1
y 8 n 12
64x 6 y 15
2x 2 +11x −8 8+2a − a 2 81 − 90y +25y 2
15
16
y 2 −6y +9 x = 5 2
1
125
x 6 49 − 9x 2
n =2 or n =4
⎛ ⎝ −∞, 1 ⎤
2⎦
f (2)=2, f (−1) = −7, f (0) = −4
a −4, 0. 5 – , 7 3 ,3, 81; b 2x 2 − x −15
−4, − 7, 0.5 – , 7 3 ,3, 81
7; c
domain: ⎡ ⎣0, ∞); range: ⎡ ⎣0, ∞)
5 ⎛ ⎝ 15 ⎞ ⎠
2
a −8 b a b −11 a b −9
a −7 b a b c a b c
a −3 b c −2 a −6 b c −4 a 6< 38

a 7 9 b 2 5 c 1 3
2|p| 6p
7
a 3u3 6u
b 2r 3
5
v 4 s 2
c 3 4
| | m3 2m 2
| |
n3
a 7z 2 b −5 2
4
c 3|m| 2m
2
a
6 b b 5
z c
4
p
3
a |a| b |x| c y 3 a x
2 b m
2 c n
2
2x 2 3x
5 a
a 5|y| x
6
c |ab| 4 a
2
3
b 2xy y 2
3
a 8m
4 4
b −4 c 3|n| 2
c 2x 2 4
5x 2
|y|
4|m| 5m
| n3 |
a 2|m| 3
4
c 2|ab| a 2
3
5 |
n 3 |
b 3c3 3
4c
d 2
3
b 3xy x 2
5
a t b 3 m c 4 r
a (10m)
1
2 b (3n)
1
5
c 3(6y)
1
4
a (3k)
1
7 b (5j)
1
4
c 8(2a)
1
3
a b c a b c a b −8
c 1 8
a b −4
3
2
2 5
a a 5 b (5ab)
3
c 1 4
5
5
a x
2 b ⎛
⎝3y ⎞ 4 c ⎛2m⎞ ⎠ ⎝ 3n ⎠
3
c ⎛7xy
⎝ z ⎞ 2
⎠
a b 1
729 c 1 8 a b 1 9 c 1
125 a −64 b − 1 64 c
a −729 b − 1
729 c
a 8x
1
5 b x
1
2
y
1 1
a u
3 b 3x 5
y
4
a −2 7x b −5xy
a
3
73b −10 5 c
3 −3 2
a 12 15 b
3 −18 4
a 144y 2
a x
3
2 b x
8 c 1 x a y
11
8 b m
2 c 1 d
1
3 2
3 a 729n
5 b a 2 3
b
3 a −2b 11
3 x
c
4
3 x −5
4 y
5y b
4 −36a 3a
a −3y b
3 37mn
a −m 3 2m b x 2 3
5x
a 27 2 b
3 −36 2
a m
2 b 5n
1
4
m
a −4 3 b
3
8 y
c 5
4 m −2 3 m
a
3
92b 22 c
3 3
a −p 3p
3 3
b 4y 4y
2
−2n 4n 2
a 288x 3 4
5 b 8y 6y
2
a 18 + 6 b
3 −2 4 −2 3 3 a −40+4 2 b
3 −3 − 18
a −66+15 7
a 41 − 14 11
1+9 21
3
b x
2 3
3
−5 x +6 b x
2 3
+4 x +3
−12−20 3 a 102 + 20 2 b 55 + 6 6 a 41 − 12 5
b 121 − 36 10

−11
−159
a 5q2
6 b 2 b a 9x2
y 2
4xy 2x
a 65
5 b 14
6 c 5x
4
a 27 b 12
3
44 c 4
35x 3
5x
4ab 3b
3
x a 49 b 90
7
4+ 6
5
⎛
⎝
p + 2 ⎞ 2
⎠
p −2
z = 3 10
4
a 125
5
4
c 24x 3
x
⎛
⎝
5 ⎛ ⎝ x − 2 ⎞ ⎠
x −2
a 5s
8 b 2 a
b −4x y a 10n 3
m
q − 10 ⎞ 2
⎠
q −10
no solution
36 c 53y2 3
3y
4
b 14
4
b −3p
q 2
a 53
3 b 8 6 c 2x x
3
a 4 b 150
2
− 3⎛ ⎝1+ 5 ⎞ ⎠
4
m = 23 3
10 ⎛ ⎝ y + 3 ⎞ ⎠
y −3
no solution
x =2,x =3 y =5,y =6 x =−6
310 c 3
25n 2
5n
x =−9 x =8 x =6
m =7 n =3 a =63
b = 311 x =3
x =− 6 5
x =4 x =9 x =5
x =0x =4 3.5
42.7
54.1
a f (6) =4b
x =0
a g(4) =5b
f (−3)
a g(4) = 2 b g(1)=−1 a h(2) = 2
b h(−5) = −3
a f (4) = 2 b f (−1) = 1 a g(16) = 3 b g(3)=2 ⎡ ⎣ 5 6 ,∞⎞ ⎠
⎛ ⎝ −∞, 4 ⎤
5⎦
(−3, ∞) (5, ∞)
(−∞, ∞) (−∞, ∞) a [−2, ∞)
b
c [0, ∞)

a [2, ∞)
b
a (−∞,∞)
b
a (−∞,∞)
b
c [0, ∞)
c (−∞,∞)
c (−∞,∞)
a 9i b 5i c 32i a 6i b 3i c 33i 62i
73i a 6+5i b 6−3i a −2−6i b 2+9i
12 + 20i 12 − 6i −11−7i
−5−10i −21−20i 9−40i
−14 −54 −12−22 3i
6 + 12 2i
4
17 + 16
17 i 2
5 + 4 5 i
3
2 − 3 2 i 4
5 − 2 5 i −i
1
a b −9 a b −1 a 2 3 b −0.1
a b −17 a −15 b a b
a b c a −2 b not real c −2 a −5 b not real c −4
a 8< 70

a | | y5 3
y b r r
2 c s 2 4
s 2
a 11 |
m 11 |
2m b 3m 2 4
5m 2
5
c 2n 5n
3
a n 10 n b q 2 3
q 2
8
c |n| n
2
a 7 |
m 3 n 5 |
3mn b 2x 2 y 2
4
c 2|xy| 2x
3
6y
a 5r 6 3
5r b 3x 4x
2
4
c 2|y| 3y
2
a 8 |
qr 3 |
3qr b 3m 3 n 3
c 3a 2 b 2 4 a
3
2n
a
3 −6 4 b a −2 b a 5+2 3 b 5− 6
a 1+3 5 b 1+ 10 a 3 4 b 2 3 c 1 3 a 5 3 b 3 5 c 1 4
a x
2 b p 3 c |q|
5m 2 3m
4
a 5r4 3r
b
s 4 3a 2 3
2a 2
b
c
4
2|c| 4c
|d|
a 2|pq| 1
3
b 2cd d 2
5
c |mn|
2
a 1 y 2 b u
2 c | v3 |
7r 2 2r
10
a 2 | p3 |
7p
|q|
c 2 | | p3 4
4p 3
| |
q3
a 3p4 p
|q|
c
5 2x 2x
b 3s 2 3
3s 2
t
b
4 22
4 |
x 3 |
6x
11
2 |
q 3 |
7
15
a 4|xy|
3
b y2 3
x
2
a 5 |
m 3 | b
3 55
4
c 3|y| 3y
2
a x b
3
y c 4
z
c |ab| 4 a
2
a 5 u b 9 v c 20 w
a x
1
7 b y
1
9 c f
1
5 a (7c)
1
3 b (12d)
1
7
1 1
a ⎛
⎝21p ⎞ 2 b ⎛
⎠ ⎝8q ⎞ 4
⎠
c 2(6b)
1
4
c 4(36r)
1
6
a b c a b c
a −6 b −6 c 1 6 a b −3 c 1 3 a b −6 c 1 6
a b −10 c 1 10
3
5 7 5
a m
2 b ⎛
⎝3y ⎞ 3 c ⎛4x⎞
⎠
⎝ 5y⎠
a u
2
5 b (6x)
5
7
4
3 c ⎛18a⎞
⎝ 5b ⎠
a b
729 1 a b 1 9 c a −27 b −
27 1 c
7
a c 8 b p 9 c 1 r
5
a y
4 b x
8 c m 1 a 81q 2 1
b a 2
b
1 1 3
7
a 8u 4 b 8p 2 4
q
a r 2 b 6s
t
a 42a b
a c
2 b 2x
3y
a 32b 7
3 m c 6
4 m a 95b 12
3 4 a c 62z
3
a −2 3 b 4pq
a −2 3 b
3 −2 5 c
4 32
a
3
73b 72 c
4 35 a a 2 2a b a 2c 3 5c b 14r 2 4
2r 2
4y 2 a −18 6 b
3 −64 9
a 72z 2 3 b 45x 2 3
2
a −42z 5
4
2z b −8y 6y
a −30 2 b
4 62
3
a 14 + 5 7 b 46 +3 3 4

a 44 − 3 11 b
4 32 + 54
4 60 + 2 3 a 30 + 18 2 b
3
x 2 3
−2 x −3
a −55 + 13 10
23 + 3 30 −439 − 2 77
3
b 2 x
2 3
+8 x +6
a 14 + 6 5 b 79 − 20 3 a 87 − 18 6
b 163 + 60 7
−227 19
3
9x 2
53 92 −5
10c 2 3 −9c 3 3 23 17q 2
37 −42 9
29 − 7 17 54 − 36 2 3
6+3 2
a 10m2
7
a 8m4
3n 4
a 4 3 b 4 3
b 3 y a 5
6r 2 b 2x
3 a 6p
q 2 b − 2a2
b
b − 2x2
2x 2 3
7y
3y 2 2ab 2a
a 56
3 b 23
9 c 25x x a 67
7 b 210
15 c 43p
3
a 121
b 3
28
11
c 3
9x
6 x
4
a 343
7
4
b 40 c 4
4 24x2 x
3
p a 25
4
a 9
−4
5 b 45
b
4
50
3
36 c 3
26a 2
3a
c
4
4 23a3 a
−2 ⎛ ⎝1+ 5 ⎞ ⎠
3 ⎛ ⎝3+ 7 ⎞ ⎠
3 ⎛ ⎝ m + 5 ⎞ ⎠
m −5
2 ⎛ ⎝ x + 6 ⎞ ⎠
⎛
⎝ r + 5 ⎞ 2
⎛
⎠
⎝ x +2 2 ⎞ 2
⎠
x −6
r −5
x −8
x =14
x =−4 m =14
v =17
m = 7 2
u =3,u =4 r =1,r =2 x =10
x =−8 x =8 x =−4
x =7 x =3 z =21
x =42 r =3 u =3
r =−2 x =1 x =−8,x =2
a =0
u = 9 4
a =4
x =0x =4 x =1x =5 x =9
8.7 4.7
a f (5)=4b a g(4)=5b g(8)=7
x =0
a F(1)=1b F(−11) = 5 a G(5)=2 6b G(2)=3 a g(6)=2b g(−2) = −2
a h(−2) = 0 3
b h(6)=2 4
a f (0)=0b f (2)=2 a g(1)=0b g(−3) = 2
⎡ ⎣ 1 3 ,∞⎞ ⎠ ⎛ ⎝ −∞, 2 ⎤
(2, ∞)
3⎦
(−∞, −3] ∪ (2, ∞) (−∞, ∞) (−∞, ∞)

⎡ ⎣ −3 8 ,∞⎞ ⎠
(−∞, ∞) a [−1, ∞)
b
a [−4, ∞)
b
a [0, ∞)
b
c [0, ∞)
a [0, ∞)
b
c [0, ∞)
a (−∞, 3]
b
c [2, ∞)
a [0, ∞)
b
c [0, ∞)
a (−∞, ∞)
b
c [0, ∞)
a (−∞, ∞)
b
c (−∞, 0]
a (−∞, ∞)
b
c (−∞, ∞)
a (−∞, ∞)
b
c (−∞, ∞)
c (−∞, ∞)
c (−∞, ∞)

a (−∞, ∞)
b
c (−∞, ∞)
a 4i b 11i c 22i a 10i b 13i c 35i 93i
82i 8+7i 14 + 2i
−2 + 2i 8+5i 7−13i
25 − 2 2i 12 + 20i −12 + 18i
−38 + + 9i 27 + 15i −7 + 24i
−5 − 12i −30 −30
−44−4i 3 −20 − 2 2i
2
25 + 11
25 i 6
13 + 13 9 i
− 12
13 − 8 13 i
4
3 − 1 3 i − 3 4 + 1 2 i
−1
a b −4 a b c a 8< 68

− 7⎛ ⎝2+ 6 ⎞ ⎠
⎛
⎝ x +2 2 ⎞ 2
⎠
2
x −8
u =3,u =4 x =−4 r =3
x =−8,x =2 x =3 64.8
a G(5)=2 6b G(2)=3 a g(1)=0b g(−3) = 2 (2, ∞)
⎛ ⎝ −∞, 10 7
⎤
⎦ a [0, ∞)
b
a (−∞, ∞)
b
c [0, ∞)
c (−∞, ∞)
82i 8+5i 23 + 14i
−8
−5 + 12i
2
25 + 11
25 i
5x 3
7
4
x
2−7 3
2x 2 3
y 9x 2 y
a 1 4 b −343
−x 2 3x 36x 4 2
7x 2 x y
3y 4
3 ⎛ ⎝2− 3 ⎞ ⎠
−12+8i 1 x =4
a [−2, ∞)
b
c [0, ∞)
82
410
5
(3x −2) 2
x 2 +18x +81 y −7 2 5n +4 2

28 63 52i
⎛ ⎝y 2 +4 ⎞ ⎠ ⎛ ⎝y 2 −5 ⎞ ⎠
−51, − 49
⎛ ⎝y −1 ⎞ ⎠ ⎛ ⎝y +1 ⎞ ⎠
x = 2 3
x = 1 2 , x =−2 1
a x
3
4 b x
2
3 c x
−2
13 inches
⎛ ⎝y −7⎞ 2
⎠
2 ⎛ ⎝x −4⎞ 2
⎠
x = 3 2 y =−3,y = 5 2
(−∞, − 4) ∪ (2, ∞)
x =4 3, x =−4 3 y =3 3, y =−3 3 x =7,x =−7
m =4,m =−4 c =2 3i, c =−2 3i c =2 6i, c =−2 6i
x =2 10, x =−2 10 y =2 7, y =−2 7
t = 83
3 , t =−83 3
x = 1 2 + 5 2 x = 1 2 − 5 2
b =−3+4 2, b =−3−4 2
r = 65
5 , r =−65 5
a =3+3 2, a =3−3 2
b =−2+2 10, b =−2−2 10
y =− 3 4 + 7 4 , y =−3 4 − 7 4
r =− 4 3 + 22i
3 , r = − 4 3 − 22i
3
a =5+2 5, a =5−2 5
t =4+
10i , t = 4− 10i
2 2
m = 7 3 , m =−1 n =− 3 4 , n =−7 4 a (a −10) 2 b ⎛ ⎝ b − 5 2⎞
⎠
a (b −2) 2 b ⎛ ⎝ n + 13 2
c ⎛ ⎝ q − 1 2
3⎞
⎠
2
⎞
⎠
c ⎛ ⎝ p + 1 2
8⎞
⎠
x =−5,x =−1 y =1,y =9
2

y =5± 10i
y =−4+3 3, y=−4−3 3
p = 5 2 + 61
2 , p = 5 2 − 61
2
z =−4+ 3i, z=−4− 3i
x =8+4 3, x =8−4 3
a =−7,a =3 b = −10, b =2
q = 7 2 + 37
2 , q = 7 2 − 37
2
c =−9,c =3
d = 11, d =−7 m =−7,m =−1 n =−2,n =8
r =− 7 3 , r =3 t =− 5 2 , t =2
y = 5 3 + 10
3 , y = 5 3 − 10
3
x =− 3 8 + 41
8 , x =−3 8 − 41
8
y =1,y = 2 3 z =1,z =− 3 2
a =−3,a =5 b =−6,b =−4
n = −2 + 5
2 6 , n = −2−2 5
6
x =−1+ 6, x =−1− 6 y =1+ 2, y =1− 2
d =
9+ 33, d = 9− 33
4
4
a b
m =
−6 + 15, m = −6 − 15
3
3
a = 1 4 + 31
4 i, a = 1 4 − 31
4 i b =− 1 5 + 19
5 i, b =−1 5 − 19
5 i
r =−5
a b
c = 2+ 3
7 , c = 2− 3
7
t = 4 5
a b
c c c
a
x = 2, x =− 2, x =2,x =−2 x = 7, x =− 7, x =2,x =−2
b
c
x =3,x =1 y =−1,y =1 x =9,x =16
x =4,x =16 x =−8,x = 343 x = 81, x = 625
x = 4 3 x =2 x = 2 5 , x = 3 4

a b a b a x =2;b (2, −7)
a x =1;b ⎛ ⎝1, −5)
⎛
⎝0, 4)
⎛
⎝0, −8)
⎛ ⎝−4, 0), ⎛ ⎝2, 0)
⎛
⎝0, −5)
⎛ ⎝−1, 0), ⎛ ⎝5, 0)
⎛
⎝0, −12)
⎛ ⎝−2, 0), ⎛ ⎝6, 0)

a
a
b g(x) = x 2 +1
f (x) = x 2
h(x) = x 2 −1
f (x) = x 2
b h(x) = x 2 +6
f (x) = x 2
h(x) = x 2 −6
f (x) = x 2

a
a
b g(x) = (x +2) 2
f (x) = x 2
b g(x) = (x +5) 2
h(x) =(x −2) 2 f (x) = x 2
h(x) =(x −5) 2
f (x) = x 2
f (x) = x 2
f (x) =−4(x +1) 2 +5
f (x) =2(x −2) 2 −5

a f (x) =3(x −1) 2 +2
b
a f (x) = −2(x −2) 2 +1
b
f (x) =(x −3) 2 −4
f (x) =(x +3) 2 −1
a
a
a
b (−4, −2)
b (−∞, 2] ∪ ⎡ ⎣6, ∞)
b (−5, −1)

a
(−∞, −4] ∪ [2, ∞)
⎡ ⎣−3, 5 ⎤ ⎦
b (−∞, 2] ∪ [8, ∞)
⎡ ⎣−1 − 2, −1+ 2 ⎤ ⎦
a
b (−∞, ∞)
⎛
⎝−∞, 4 − 2 ⎞ ⎠ ∪ ⎛ ⎝4+ 2,∞ ⎞ ⎠
a (−∞, ∞)
b
a =±7 r =±26 u = ± 10 3
m =±3 x =±6 x =±5i
x =±37i x =±9 a =±25
p =± 47
7
y =± 410
u = 14, u =−2
5
m =6±2 5
r = 1 2 ± 2
3 y =− 2 3 ± 22
9
a =7±5 2 x =−3±2 2
c =− 1 5 ± 33
5 i
x = 3 4 ± 7 2 i m =2±2 2 x =3±2 3
x =− 3 5 , x = 9 5 x =− 7 6 , x = 11 6
a =4±2 7
p = 1 3 ± 7
3
w =1,w = 5 3
m =4±2 3 x =−3,x =−7
x =6±2i
r =±4
a =±32
m =±23i u =7±6 2
c =± 56
6
a (m −12) 2 b ⎛ ⎝ x − 11 2
c ⎛ ⎝ p − 1 2
6⎞
⎠
2
⎞
⎠

a ⎛ ⎝p −11 ⎞ 2
⎠ b ⎛ ⎝ y + 5 2
2⎞
⎠
c ⎛ ⎝ m + 1 2
5⎞
⎠
u =−3,u =1 x =−1,x =21
m =−2±2 10i r =−3± 2i a =5±2 5
x =− 5 2 ± 33
2
u =1,u =13 r =−2,r =6
v = 9 2 ± 89
2
x =5± 30 x =−7,x =3
x =−5,x =−1 m = −11, m =1 n =−1± 14
c =−2,c = 3 2 x =−5,x = 3 2 p =− 7 4 ± 161
4
x =
10 3 ± 191
10 i
m =−1,m = 3 4
p = 1 2 , p =3 p =−4, p =−3 r =−3,r =11
u =
−7 ± 73
6
a = 3± 3
2
x =−4±2 5
y =−2,y = 1 3 x =− 3 4 ± 15
4 i x = 3 8 ± 7 8 i
v =2± 13
b =
−2 ± 22
6
a no real solutions b 1
c 2
a Quadratic Formula
b square root
c factor
x =± 15, x =± 2i
x =−1,x =12
y = 3 ± 193
2
m =−1,m = 3 4
c =− 3 4 q =− 3 5
a 1 b no real solutions
c 2
a factor
b square root
c Quadratic Formula
x =± 3, x =±2
x =±1,x = ± 6
2
x =± 3, x =± 2 2
x =− 5 3 , x =0 x =0,x =± 3
x =± 22
2 , x =± 7 x =25 x =4
x = 1 4 x = 1 25 , x = 9 4
x =8,x = −216
x =1,x =49
x =−1,x = −512
x = 27 8 , x =−64 27 x = 27
512 , x = 125
x =−2,x =− 3 5 x =−2,x = 4 3

a b
a b a x =−4 b ⎛ ⎝−4, −17) a x =1 b ⎛ ⎝1, 6)
⎛
⎝0, 6);
⎛ ⎝−1, 0), ⎛ ⎝−6, 0)
⎛
⎝0, 13);
⎛
⎝0, 12);
⎛ ⎝−2, 0), ⎛ ⎝−6, 0)
⎛ ⎝ 5 2 ,0⎞ ⎠
⎛
⎝0, 25);
⎛
⎝0, −19);
⎛
⎝0, −9);
⎛ ⎝−3, 0)

− 9 8
x =− 1 4 .

a
b g(x) = x 2 +4
f (x) = x 2
h(x) = x 2 −4
f (x) = x 2
a
b g(x) = (x −3) 2
f (x) = x 2
h(x) =(x +3) 2
f (x) = x 2

f (x) =−3(x +2) 2 +7 f (x) =3(x +1) 2 −4 a f (x) =(x +3) 2 −4
b

a f (x) =(x +2) 2 −1
b
a f (x) =(x −3) 2 +6
b
a f (x) =−(x −4) 2 +0
b
a f (x) =−(x +2) 2 +6
b
a f (x) =5(x −1) 2 +3
b
a f (x) =2(x −1) 2 −1
b
a f (x) =−2(x −2) 2 −2
b
a f (x) =2(x +1) 2 +4
b
a f (x) =−(x −1) 2 −3
b
c e d

g f (x) =(x +1) 2 −5 f (x) =2(x −1) 2 −3
a
a
b (−∞, −5) ∪ (−1, ∞)
a
a
b [−3, −1]
(−∞, −4] ∪ [1, ∞)
b (−∞, −6] ∪ [3, ∞)
b [−3, 4]
(2, 5) (−∞, −5) ∪ (−3, ∞) ⎡ ⎣2− 2,2+ 2 ⎤ ⎦
⎛ ⎝−∞, 5 − 6 ⎞ ⎠ ∪ ⎛ ⎝5+ 6,∞ ⎞ ⎠
⎡ ⎣ 2 3 , 5 ⎤
2⎦
⎡ ⎣ −1 2 , 4 ⎤ ⎦
(−∞, ∞). (−∞, ∞).
y =±12 a =±5 r =±42i
w =±53 p =1, p =9
x = 1 4 ± 4
3
n =4±10 2 n =−5±2 3 (x +11) 2
⎛ ⎝ a − 3 2
⎞
d = −13, −1 m =−3±10i
2⎠
v =7±3 2
m =−9,−1
a = 3 2 ± 41
2

u =−6±3 2 p =0,6
c =− 1 3 ± 27
3
r =−6,7
a = 5 12 ± 23
12 i u =5± 22
c =− 1 2
x =± 2, ±2 3
x = 64, 216
y =− 1 2 ,2
x = 3 2 ± 1 2 i x = 1 4 ,1
v =− 5 4 , 1 m =
a b c d
x = ± 1, ± 1 2
x =−2, 4
3
−4 ± 10
3
p = 4± 6
5
a b
c
x =16
a b x = 2; (2, −7)
y : (0, −46)
x : none
y : (0, 64)
x : (−8, 0)
y : (0, 15)
x : (3, 0), (5, 0)

f (x) =2(x −1) 2 −6
a f (x) =3(x −1) 2 −4
b
a f (x) =2(x +1) 2 +4
b
a f (x) =−3(x +2) 2 +7
b
f (x) =(x +1) 2 −5
a
a
(2, 4)
b (−∞, −2) ∪ (3, ∞)
⎡ ⎣3− 5,3+ 5 ⎤ ⎦
b [−2, 1]

w =−2,w =−8
m =1,m = 3 2 y = 2 3
y =1,y = −27 a b x =−4
c (−4, 0) d
y: (0, −16); x: (−4, 0)
e 0
x =−4
⎛ ⎝ −∞, − 5 ⎞
2⎠ ∪ (2, ∞)
f (x) =2(x −1) 2 −6
a f (4) =5 b
g ⎛ ⎝f (4) ⎞ ⎠ =32
x =− 2 3 y +4
x
x a 1 b 1 a 1 2 b 3
x =9, x =−9
a 1 b a
1
9
1
⎛
⎝ x 2 y ⎞ 3
⎠
x =7
2.565
x =4, x =−4 x =2, x =3 x =−5,x =1
a 15x +1 b 15x −9
c 15x 2 −7x −2
a b c
a b
a 24x −23 b 24x −23
c 24x 2 −38x +15
a
b
a
b
a b c
a
b
{(4, 0), (7, 1), (10, 2), (13, 3)}.
{4, 7, 10, 13}.
{0, 1, 2, 3}.

{(4, −1), (1, −2), (0, −3), (2, −4)}.
{0, 1, 2, 4}.
{−4, −3, −2, −1}.
g( f (x)) = x,
g( f (x)) = x,
f (g(x)) = x,
f (g(x)) = x,
f −1 (x) = x −5
8 f −1 (x) = x5 +2
3
f −1 (x) = x +3
5
f −1 (x) = x4 +7
6
x =2
x =4 x =−1,x =2 x =−2,x =3
a $22,332.96
b $22,362.49 c $22,377.37
a b
c
a log 3 9=2
b log 7
7 = 1 2 c log 1
3
1
27
= x
a log 4 64 = 3
3
b log 4 4
= 1 3 c log 1
2
1
32
= x

a 64 = 4
3
a 27 = 3
3 b 1=x
0
b 1=x
0 c
100 1 =10−2 c
10 1 =10−1
a
x =9b x = 243 c x =3
a
b 1 2 c −5
a x =8b x = 125 c x =2
a b 1 3 c −2
a
a =11
b x = e
7
a
a =4
b x = e
9
a
x =13
b x =2
a
x =6
b x =1
a b a b a b
a b
a 1 + log 3 x
b 3 + log 2 x + log 2 y
a 1 + log 9 x
b 3 + log 3 x + log 3 y
a log 4 3−1 b log x −3 a log 2 5−2 b 1 − logy a 4log 7 5 b 100 · log x
a 7log 2 3 b 20 · log x log 2 5 + 4log 2 x + 2log 2 y log 3 7 + 5log 3 x + 3log 3 y
1⎛
5⎝ 4log 4 x − 1 2 − 3log 4 y − 2log 4 z⎞ ⎠
1⎛
3
⎝2log 3 x − log 3 5 − log 3 y − log 3 z ⎞ ⎠
log 2
5x y
log 3
6 xy log 2 x 3 (x −1) 2 logx 2 (x +1) 2
3.402 2.379 x =6
x =4 x =4 x =8
x =3 x =8
x = log43
log7 ≈ 1.933

x = log98
log8 ≈ 2.205 x = ln9 + 2 ≈ 4.197
x = ln5
2 ≈ 0.805
r ≈ 9.3% r ≈ 11.9%
a 8x +23 b 8x +11 c
8x 2 +26x +15
a 2x 2 +3 b 4x 2 −4x +3
c 2x 3 − x 2 +4x −2
a 24x +1 b 24x −19
c 24x 2 −14x −5
a b c
a 6x 2 −9x b 18x 2 −9x
c 6x 3 −9x 2
a b c
a b
a
b
{(−3, −2), (−1, −1), (1, 0), (3, 1)}.
{−3, −1, 1, 3}.
{−2, −1, 0, 1}.
{(1, 2), (2, 4), (3, 6), (4, 8)}.
{1, 2, 3, 4}.
{2, 4, 6, 8}.
{(−2, 0), (3, 1), (7, 2), (12, 3)}.
{−2,3,7,12}.
{0, 1, 2, 3}.
g( f (x)) = x,
f (g(x)) = x,
g( f (x)) = x,
f (g(x)) = x,
x).
g( f (x)) = x,
g( f (x)) = x,
f (g(x)) = x,
f (g(x)) = x,
f −1 (x) =x +12 f −1 (x) =
9
x
f −1 (x) =6x f −1 (x) = x +7
6 f −1 (x) = x −2
−5
f −1 (x) = x −6 f −1 3
(x) = x +4
f −1 (x) = 1 x −2
f −1 (x) =x 2 +2 x ≥0 f −1 (x) =x 3 +3
f −1 (x) = x5 −5
−3
f −1 (x) = x4 +5
9
x ≥0

x =4
x =−1 x =−1,x =1 x =1
x =−1 x =2 x =−1,x =2
f a e
a $7,387.28 b $7,434.57
c $7,459.12
$36,945.28

log 2 32 = 5 log 5 125 = 3
log
100 1 3
log x 6 = 1 5
log
3
17 17 = x log 1
1 =4
3 81
log 4
1
64
=−3 ln x =3 64 = 2 6
32 = x 5 1=7 0 9=9 1
1,000 = 10 3 43 = e x x =11
x =4 x = 125
x = 1
243
x =2 x =−2
1
3
−2
−3
−2
a =9 a =3
3
a =2 3
x = e 4 x =5
x =17 x =6 x =3
x =−5 5, x =5 5 x =−5,x =5
a b a b

a b −4 a 3 b −1 a b
log 5 8 + log 5 y 4 + log 3 x + log 3 y 3 + logy
log 6 5−1 3 − log 5 x 4 − logy
4 − ln16 5log 2 x −3log x
1
3 log 5 x 3 4ln x log 2 3 + 5log 2 x + 3log 2 y
1
4 log 5 21 + 3log 5 y log 5 4 + log 5 a + 3log 5 b
2
3 log 3 x − 3 − 4log 3 y
+ 4log 5 c − 2log 5 d
1
2 log 3 ⎛ ⎝3x +2y 2⎞ ⎠ − log 3 5 − 2log 3 z
1⎛
3
⎝log 5 3 + 2log 5 x − log 5 4
− 3log 5 y − log 5 z ⎞ ⎠
log 5 2 x −1
log 2 5 xy
log 3 x 6 y 9
ln x3 y 4
z 2
log(2x +3) 2 · x +1 2.379 1.674
5.542
x =7 x =4 x =1, x =3
x =8 x =3 x =20
x =3 x =6
x = 5 3
x = log74
log2 ≈ 6.209
x = log8
log 1 ≈ −1.893
3
x = log112
log4 ≈ 3.404 x = ln8 ≈ 2.079
x = ln3 − 2 ≈ −0.901
x = ln16
3 ≈ 0.924
x = ln6 ≈ 1.792 x = ln8 + 1 ≈ 3.079 x =5
x =−4,x =5 a =3 x = e 9
x =7 x =3 x =2
x =6 x =5
x = log10
log 1 ≈ −3.322
2
x = ln7 − 5 ≈ −3.054 6.9%
a 4x 2 +12x b 16x 2 +12x a −123 b c
c 4x 3 +12x 2
a
b
{(10, −3), (5, −2), (2, −1), (1, 0)}.
{1, 2, 5, 10}.
{−3, −2, −1, 0}.
g( f (x)) = x,
f (g(x)) = x,

f −1 (x) = x +11
6
f −1 (x) = 1 x −5
x =−2,x =2 x =−1 x =−3,x =5
$163,323.40
log 1,000 1 ln16 = y 100000 = 10 5 x =5
x =4
x = e −3 x =8
a b −9 a b
4 + logm
log 5 8 + 2log 5 a + 6log 5 b
+ log 5 c − 3log 5 d
5 − ln2
1⎛
3
⎝log 6 7 + 2log 6 x − 1 − 3log 6 y
− 5log 6 z ⎞ ⎠
1
7 log 4 z
log 3 x 3 y 7
4
y
x =4
log
(y −3) 2
x =3
x = log101
log2 ≈ 6.658
x = log7
log 1 ≈ −1.771
3
x = ln15 + 4 ≈ 6.708

a 48x −17 b 48x +5
c 48x 2 −10x −3
a b
x =5 a $31,250.74 b $31,302.29
f −1 5
(x) = x +9
343 = 7 3
c $31,328.32
2 + log 5 a + log 5 b
1⎛
4
⎝log 2 5 + 3log 2 x − 4 − 2log 2 y
log
6
x
(x +5) 3
− 7log 2 z ⎞ ⎠
x =6 x = ln8 + 4 ≈ 6.079
20 inches (x −9) 2 x =6±4 3
x =3± 3 y =3(x −1) 2 +2
(x −4) 2 =8 y =2(x −3) 2 −4

x =±23 x 2 −8x +16
(3, 2)
⎛ ⎝ 2, 3 ⎞
2⎠
(1, 3)
d =5 d =10 d =15
d =13 d = 130, d ≈ 11.4 d = 2, d ≈ 1.4
x 2 + y 2 =36
x 2 + y 2 =64 (x −2) 2 + ⎛ ⎝y +4 ⎞ ⎠2 =49 (x +3) 2 + ⎛ ⎝y +5 ⎞ ⎠2 =81
(x −2) 2 + ⎛ ⎝y −1 ⎞ ⎠2 =25 (x −7) 2 + ⎛ ⎝y −1 ⎞ ⎠2 = 100 a
(3, −4)
b

a
(3, 1)
b
a
(0, 0)
b
a
(0, 0)
b
a
(3, 4)
b
a
(−3, 1)
b
a
(1, 0)
b
a
(0, 6)
b

a y =2(x +1) 2 +3
b
a y =−2(x −2) 2 +1
b
a x =3(y +1) 2 +4
b
a x =−4(y +2) 2 +4
b

y =− 1 20 (x −20)2 +20 y =− 1 5 (x −5)2 +5
x 2
4 + y2
25 =1 x 2
9 + y2
4 =1

a
b
(x +1)2
6
+ ⎛ ⎝y −4 ⎞ 2
⎠
=1
9
a
b
(x −2)2
4
+ ⎛ ⎝y −3 ⎞ 2
⎠
=1
16
x 2
625 + y2
600 =1
x 2
1225 + y2
1000 =1

a
b
(x +1)2
16
− ⎛ ⎝y −2 ⎞ 2
⎠
=1
9
a
b
(x +3)2
25
− ⎛ ⎝y +1 ⎞ 2
⎠
=1
16
a b c
d
a b c
d
⎛ ⎝ −4 5 , 6 ⎞
5⎠ , (0, 2)
(−1, 0), (0, 1), (0, −1)
(−3, 0), (3, 0), ⎛ ⎝−2 2, −1 ⎞ ⎠, ⎛ ⎝22,−1 ⎞ ⎠
⎛ ⎝ 4 9 , − 2 ⎞
3⎠ , (1, 1)
(−3, −4), (−3, 4), (3, −4), (3, 4)
(−2, 0), (2, 0) −18
d =5
d =3 5, d ≈ 6.7 d =2 17, d ≈ 8.2

a (2, −4)
b
a
b
⎛
⎝ 31 2 ,−11 ⎞
2⎠
x 2 + y 2 =49
x 2 + y 2 =2 (x −3) 2 + ⎛ ⎝y −5 ⎞ ⎠2 =1 (x − 1.5) 2 + ⎛ ⎝y + 3.5 ⎞ ⎠2 = 6.25
(x −3) 2 + ⎛ ⎝y +2 ⎞ ⎠2 =64 (x −4) 2 + ⎛ ⎝y −4 ⎞ ⎠2 =8 a
(−5, −3)
b
a
(4, −2)
b
a
(0, −2)
b
a
(1.5, −2.5) 0.5.
b

a
(0, 0)
b
a
(0, 0)
b
a (−1, −3),
b
a (2, −5),
a (0, −3),
a (−2, 0),
b
b
b

a y =−(x −1) 2 −3
b
a y =−2(x +1) 2 −3
b

a x = ⎛ ⎝y +2 ⎞2 ⎠ −9
b
a x =−2 ⎛ ⎝y +3 ⎞ ⎠2 +2
b
a b d
y =−
15 1 (x −15)2 +15 y =−
10 1 (x −30)2 +90
x 2
9 + y2
25 =1 x 2
9 + y2
16 =1

a
b
(x −2)2
9
+ ⎛ ⎝y −3 ⎞ 2
⎠
=1
25
a y2
4
b
+
(x −3)2
25
=1
x 2
400 + y2
300 =1 x 2
2500 + y2
1275 =1

a
b
(x −1)2
4
− ⎛ ⎝y −1 ⎞ 2
⎠
=1
9
a ⎛ ⎝y −2 ⎞ 2
⎠ (x −1)2
− =1
9 9
b
a ⎛ ⎝y +1 ⎞ 2
⎠ (x +2)2
− =1
1 9
b
a b c
d

(−1, 0), (0, 3) (2, 0)
(12, −5), (12, 5) (0, −4), (1, −3)
(3, 4), (5, 0) (0, −4), ⎛ ⎝−7,3 ⎞ ⎠, ⎛ ⎝ 7, 3 ⎞ ⎠ (0, −2), ⎛ ⎝−3,1 ⎞ ⎠, ⎛ ⎝ 3, 1 ⎞ ⎠
(−2, 0), ⎛ ⎝1, − 3 ⎞ ⎠, ⎛ ⎝1, 3 ⎞ ⎠
(−2, −4), (−2, 4), (2, −4), (2, 4)
⎛ ⎝−3,0 ⎞ ⎠, ⎛ ⎝ 3, 0 ⎞ ⎠
(−2, −3), (−2, 3), (2, −3), (2, 3)
(−4, 0), (4, 0)
(−1, −3), (−1, 3), (1, −3), (1, 3)
−3 −7 −8 −6 −4 −6
−4
d =3 d = 17, d ≈ 4.1 (4, 4)
⎛ ⎝ 3 2 , − 7 ⎞
2⎠
x 2 + y 2 =7 (x +2) 2 + ⎛ ⎝y +5 ⎞ ⎠2 =49

(x −2) 2 + ⎛ ⎝y −2 ⎞ ⎠2 =8 a (0, 0)
b
a (−2, −5)
b
a (0, 2)
b
a y =2(x −1) 2 −4
b
a y =−(x −6) 2 +1
b

a x = ⎛ ⎝y +2 ⎞2 ⎠ +1
b
a x =−2 ⎛ ⎝y −1 ⎞ ⎠2 +2
b
y =− 1 9 x2 + 10 3 x
x 2
36 + y2
64 =1
a
b
(x −3)2
4
+ ⎛ ⎝y −7 ⎞ 2
⎠
=1
25
a x 2
9 + ⎛ ⎝y −7 ⎞ 2
⎠
=1
4
b

a
b
(x +1)2
16
− ⎛ ⎝y −3 ⎞ 2
⎠
=1
4
a ⎛ ⎝y −1 ⎞ 2
⎠ (x −1)2
− =1
16 4
b
a b c
d
(−1, 4)
⎛ ⎝−7,3 ⎞ ⎠, ⎛ ⎝ 7, 3 ⎞ ⎠ (−3, 0), (0, −2), (0, 2)
−3 −4
(−7, −7)
x 2 + y 2 = 121 (x +2) 2 + ⎛ ⎝y −3 ⎞ ⎠2 =52

a
b
a
b
a
b
a
b
a
b (x +5) 2 + ⎛ ⎝y +3 ⎞ ⎠2 =4
c
a
b
c
(x −2)2
25
− ⎛ ⎝y +1 ⎞ 2
⎠
=1
9
(3, 0), (−3, 0)
x 2
2025 + y2
1400 =1
−1, 1, −1, 1
(5, 2)
3
4 a b
16
1
9x 2 +30x +25 x 2 −2xy + y 2
a b c

−1,2,5,8,11 −3, −1, 1, 3, 5 7, 13, 31, 85, 247
−3, −1, 3, 11, 27 −1, 4, −9, 16, −25 1, −8, 27, −64, 125
a n =3n a n =5n a n = (−1) n 3 n
a n = (−1) n +1 n 2 a n = 1 2 n a n = 1 n 2
2, 1, 1 3 , 1 , 1
12 60 3, 3 2 , 1 2 , 1 8 , 1
40
1
2 , 1 3 , 1 4 , 1 5 , 1 6
16
3
1
2 , 1 6 , 1
12
, 1
20
, 1
30
5
∑
n =1
5
5
∑ 1
n =1n ∑ (−1) n +1 n 2 5
∑ (−1) n 2n
n =1
n =1
a
a
7, 3, −1, −5, −9, …
d =11 b
b
d =−6
c
d =2.
c
d =−5.
11, 3, −5, −13, −21, … −106
a 11 =2.
a n =−3n + 35.
a 1 =8, d =−3.
a n =−3n + 11.
a 19 = −55.
a n =−4n + 21.
1
2 n
a 1 =5, d =4.
a n =4n +1.
a
r =3.b
r = 1 4 . c
a
b
r =2.c
7, −21, 63, −189, 567 6, −24, 96, −384, 1536
r = 1 2 .
1
6,561
a 11 = 7,168.
a n =7(2) n −1 .
1
a 9 = 39,366.
16,384
a n =6(3) n −1 .
3,145,725 10,460,353,200 393,204
10,230
256
3
4
9
8
9
$10,000
$3, 333.33 $88,868.36 $698,201.57

x 5 +5x 4 y +10x 3 y 2 +10x 2 y 3
+5xy 4 + y 5
x 6 +6x 5 +15x 4 +20x 3 +15x 2
+6x +1
p 7 +7p 6 q +21p 5 q 2 +35p 4 q 3 x 4 +8x 3 +24x 2 +32x +16
+35p 3 q 4 +21p 2 q 5 +7pq 6 + q 7
16x 4 −96x 3 + 216x 2 − 216x +81
64x 6 − 192x 5 + 240x 4 − 160x 3
+60x 2 −12x +1
a b c d a b c d
x 5 +5x 4 y +10x 3 y 2 +10x 2 y 3
x 5 −15x 4 +90x 3 − 270x 2
m 6 +6m 5 n +15m 4 n 2 +20m 3 n 3 +405x − 243
+15m 2 n 4 +6mn 5 + n 6
+5xy 4 + y 5
y 6 −6y 5 +15y 4 −20y 3 +15y 2
−6y +1
256x 4 − 768x 3 y + 864x 2 y 2 15x 4 y 2
−720x 2 y 3 + 240xy 4 −32y 5
243x 5 − 810x 4 y + 1080x 3 y 2
−432xy 3 +81y 4
70a 4 b 4
−5, −3, −1, 1, 3 4, 7, 10, 13, 16 5, 7, 11, 19, 35
1, 5, 21, 73, 233
2, 1, 8 9 ,1,32 25 1, 3 2 , 5 4 , 7 8 , 9
16
−2, 4, −6, 8, −10 1, −4, 9, −16, 25
1, − 1 4 , 1 9 , − 16 1 , 1
25
a n =8n a n = n +5 a n = e n +2
a n = (−1) n 5n a n = (−1) n n 3 a n = (−1) n 2n
a n = 1 4 n a n =−
n +1
n
a n =− 5 2 n
4, 2, 2 3 , 1 6 , 1
30
3, 6, 18, 72, 360 2, 24, 720, 40320, 3628800
1, 1 2 , 1 3 , 1 4 , 1 5 1, 1 2 , 2 3 , 3 2 , 24 5 2, 3 2 , 8 3 , 15 2 , 144
5
1+4+9+16+25=55
2+4+8+16=30
5+7+9+11+13+15=60
4
1 + 4 1 + 4 2 + 4 6 = 32 3 =102 3
2 + 6 + 12 + 20 + 30 = 70 1
2 + 2 3 + 3 4 + 4 5 + 5 6 = 71
20
5
5
5
∑ 1
n =13 ∑ 1
n =1n ∑ 2
n
n =1
5
∑ (−1) n +1 10
3n ∑ (−1) n 7
2n ∑ (2n + 12)
n =1
n =1
n =1
d =8.
11, 18, 25, 32, 39
d =−3.
−7, −3, 1, 5, 9 14, 5, −4, −13, −22 163
131 −79 a 20 = −34.
a n =−2n +6.

a 11 = 59.
a n =5n +4.
a 1 = 21, d =−8.
a n =−8n + 29.
a 8 = −13.
a n =−5n + 27.
a 1 = −15, d =3.
a n =3n − 18.
a 1 = 11, d =3.
a n =3n +8.
−1, 065
−3,850
r =4.
r = 1 2 .
r = 1 2 .
d =5.
r =−2.
r = 1 2 .
4, 12, 36, 108, 324
−4, 8, −16, 32, −64
27, 9, 3, 1, 1 3
472,392 0.0001
a 9 = 2,304.
a
a n =9(2) n −1 15 =− 2
19, 683 . a 10 = 0.000000001.
.
a n = (0.1) n −1 .
a n = −486 ⎛ n −1
⎝ −1 ⎞
3⎠
.
57,395,624 −65,538
7,174,453
59,049 ≈ 121.5
65,534 4088
3
2
5
11
9
2
1
3
a $6666.67 b $4000 c
$15,000 c $7500
29,524
6561 ≈ 4.5
r ≥1
7
9
$295,581.88
$14,234.10
a 8 +8a 7 b +28a 6 b 2 +56a 5 b 3
+70a 4 b 4 +56a 3 b 5 +28a 2 b 6
+8ab 7 + b 8
x 3 +15x 2 +75x + 125
243x 5 − 405x 4 + 270x 3 −90x 2
+15x −1
p 9 +9p 8 q +36p 7 q 2 +84p 6 q 3 a 6 −6a 5 b +15a 4 b 2 −20a 3 b 3
+126p 5 q 4 + 126p 4 q 5 +84p 3 q 6 +15a 2 b 4 −6ab 5 + b 6
+36p 2 q 7 +9pq 8 + q 9
y 7 +7y 6 +21y 5 +35y 4 +35y 3
+21y 2 +7y +1
27x 3 − 135x 2 + 225x − 125
z 6 −12z 5 +60z 4 − 160z 3 + 240z 2
−192z +64
27x 3 + 135x 2 y + 225xy 2 + 125y 3
a b c d a b c d
m 5 +5m 4 n +10m 3 n 2 +10m 2 n 3
s 7 +7s 6 t +21s 5 t 2 +35s 4 t 3
+35s 3 t 4 +21s 2 t 5 +7st 6 + t 7
y 4 −12y 3 +54y 2 − 108y +81
+5mn 4 + n 5
q 3 −12q 2 +48q −64

625x 4 − 1000x 3 y + 600x 2 y 2
−160xy 3 +16y 4
126a 5 b 4
243x 5 + 1620x 4 y + 4320x 3 y 2
+5760x 2 y 3 + 3840xy 4 + 1024y 5
462x 5 y 6
7, 13, 31, 85, 247
3
4 , 5 , 7 , 9
16 64 256 , 11
1024
a n = e n −4 a n =−
n +2
n
−3+(−1)+1+3+5
+7 + 9 = 21
5
∑ 4
n
n =1
a n =9n
1
6 , 1 , 1 , 1 , 1
12 20 30 42
4+4+2+ 2 3 + 1 6 = 65 5
6 ∑ (−1) n 1
n =1 3 n
5, 8, 11, 14, 17
d =6.
−13, −7, −1, 5, 11 −129 a 18 = 103.
a n =7n − 23.
a 1 =1, d =4.
a n =4n −3.
−1,095 585
4,850 980
r =−2.
128, 32, 8, 2, 1 2
1,536
a 12 = −25, 165, 824.
a n = 6(−4) n −1 .
3906.25
189
8 = 23.625
$1,634,421.27
343
6 ≈ 57.167
x 4 −4x 3 y +6x 2 y 2 −4xy 3 + y 4
4
11
32y 5 − 240y 4 + 720y 3 − 1080y 2
+810y − 243
a b c d a b c d
t 9 −9t 8 +36t 7 −84t 6 + 126t 5
256x 4 + 768x 3 y + 864x 2 y 2
+432xy 3 +81y 4 84a 3 b 6
1
4 , 1 5 , 1 6 , 1 7 , 1 8
a 23 = 59.
a n =3n − 10.
−4 + 16 − 64 + 256 = 204
1,325 3,260
5,242,880 797,162
r = 1 3 .
5
6
$1,409,344.19
−126t 4 +84t 3 −36t 2 +9t −1
−13, −10, −7, −4, −1
a b c d

3×3