Algebra Cheat Sheet (Reduced) - Pauls Online Math Notes

Algebra Cheat Sheet
Basic Properties & Facts
Arithmetic Operations
Properties of Inequalities
æbö
ab If a < b thena+ c< b+ c and a- c< b-c
ab+ ac= ab ( + c)
aç ÷ =
è c ø c
a b
If a < b and c > 0 then ac< bc and <
æaö
c c
ç ÷
èb
ø a a ac
a b
= =
If a < b and c< 0 then ac> bc and >
c bc æb
ö b
c c
ç
c
÷
è ø
Properties of Absolute Value
a c ad + bc a c ad -bc
ìa
if a³
0
+ = - =
a =í
b d bd b d bd
î - a if a < 0
a-b b- a a+
b a b a ³ 0
- a = a
= = +
c-d d -c c c c
a a
ab = ab
=
æaö
b b
ab+ ac
ç
b
÷
ad
= b+ c, a ¹ 0
è ø
=
a+ b £ a + b Triangle Inequality
a
æ c ö bc
ç
d
÷
è ø
Distance Formula
Exponent Properties
If P
n
1
= ( x1,
y1)
and P2 = ( x2,
y2)
are two
n m n+ m a n-m
1
aa = a = a = points the distance between them is
m
m-n
a a
n
( )
m
nm
0
a a a a
( ab)
a
-n
= = 1, ¹ 0
n n n æaö
a
= ab
ç ÷ =
èbø
b
1 1 n
= = a
n
-n
a
a
-n n n
n
n
m
n
n
n
n
1
m
( ) ( ) 1
m n
æaö æbö
b
ç ÷ = ç ÷ = a = a = a
èbø è aø
a
Properties of Radicals
1
n n
n n n
m n nm n
n
n
a = a ab = a b
a
a = a =
b
n
a = a,if
n is odd
n
a = a ,if n is even
n
n
a
b
( , ) = ( - ) + ( - )
2 2
1 2 2 1 2 1
d P P x x y y
Complex Numbers
i = - i =- - a = i a a³
( a+ bi) + ( c+ di) = a+ c+ ( b+
d)
i
( a+ bi) -( c+ di) = a- c+ ( b-d)
i
( + )( + ) = - + ( + )
2 2
( a+ bi)( a- bi)
= a + b
( )
2
1 1 , 0
a bi c di ac bd ad bc i
a+ bi = a + b
2 2
a+ bi = a-bi
( )( )
a+ bi a+ bi = a+
bi
Complex Modulus
Complex Conjugate
2
Logarithms and Log Properties
Definition
y
y= log x is equivalent to x=
b
b
Example
3
log 125= 3 because 5 = 125
5
Special Logarithms
lnx=
log x natural log
e
logx
= log10
x common log
where e= 2.718281828K
Factoring Formulas
2 2
x - a = x+ a x-a
( )( )
( )
2 2
2
x + 2ax+ a = x+
a
2
2 ( )
( ) ( )( )
3
3 3 ( )
- + = -
2 2
x ax a x a
+ + + = + +
2
x a b x ab x a x b
3 2 2 3
x + ax + ax+ a = x+
a
3 2 2 3
3
x - 3ax + 3ax- a = x-a
( )
( )( )
( )( )
( )( )
x + a = x+ a x - ax+
a
3 3 2 2
x - a = x- a x + ax+
a
3 3 2 2
x - a = x - a x + a
2n 2n n n n n
If n is odd then,
x - a = x- a x + ax + L+
a
x
-1 -2 -1
( )( )
n n n n n
n
+ a
n
n-1 n-2 2 n-3 n-1
( x a)( x ax ax L a )
= + - + - +
2
Solve 2x
-6x- 10=
0
2
(1) Divide by the coefficient of the x
2
x -3x- 5=
0
(2) Move the constant to the other side.
2
x - 3x=
5
(3) Take half the coefficient of x, square
it and add it to both sides
2 2
2 æ 3ö æ 3ö
9 29
x - 3x+ ç - ÷ = 5+ ç - ÷ = 5+ =
è 2ø è 2ø
4 4
Logarithm Properties
log b= 1 log 1=
0
log
log
b
x
logb
x
bb = x b = x
b
r
( )
x = rlog
x
( )
log xy = log x+
log y
b b b
æ xö logbç ÷ = logb x - logb
y
è yø
The domain of log b
x is x> 0
Factoring and Solving
Quadratic Formula
2
Solve ax + bx+ c= 0 , a ¹ 0
If b
If b
If b
2
2
2
b
b
2
- b± b -4ac
x =
2a
- 4ac
> 0 - Two real unequal solns.
- 4ac
= 0 - Repeated real solution.
- 4ac
< 0 - Two complex solutions.
Square Root Property
2
If x = p then x=±
p
Absolute Value Equations/Inequalities
If b is a positive number
p = b Þ p=- b or p=
b
p < b Þ - b< p<
b
p > b Þ p
b
Completing the Square
(4) Factor the left side
æ 3ö
29
ç x- ÷ =
è 2ø
4
(5) Use Square Root Property
3 29 29
x- =± =±
2 4 2
(6) Solve for x
2
3 29
x= ±
2 2
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu.
© 2005 Paul Dawkins
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu.
© 2005 Paul Dawkins

Functions and Graphs
Constant Function
Parabola/Quadratic Function
y= a or f ( x)
= a
2
x= ay + by+ c
2
g( y)
= ay + by+
c
Graph is a horizontal line passing
through the point ( 0,a ) .
The graph is a parabola that opens right
if a > 0 or left if a < 0 and has a vertex
Line/Linear Function
æ æ b ö b ö
at ç gç - ,
y= mx+ b or f ( x)
= mx+
b
2a
÷ -
2a
÷
è è ø ø .
Graph is a line with point ( 0,b ) and
Circle
slope m.
( x- h) 2 + ( y- k)
2 = r
2
Slope
Slope of the line containing the two
Graph is a circle with radius r and center
( hk , ) .
points ( x1,
y
1)
and ( x2,
y
2)
is
y2 - y1
rise
Ellipse
m= = x
2
- x
1
run
( x-h) 2 ( y-k)
2
+ = 1
2 2
Slope – intercept form
a b
The equation of the line with slope m Graph is an ellipse with center ( hk , )
and y-intercept ( 0,b ) is
with vertices a units right/left from the
y = mx+
b
Point – Slope form
center and vertices b units up/down from
the center.
The equation of the line with slope m
and passing through the point ( x1,
y
1)
is Hyperbola
y= y1+ m( x-
x1)
( x-h) 2 ( y-k)
2
- = 1
2 2
a b
Graph is a hyperbola that opens left and
Parabola/Quadratic Function
2 2 right, has a center at
y= a( x- h) + k f ( x) = a( x- h)
( hk , ) , vertices a
+ k
units left/right of center and asymptotes
b
The graph is a parabola that opens up if that pass through center with slope ± .
a > 0 or down if a < 0 and has a vertex
a
Hyperbola
at ( hk , ) .
( y-k) 2 ( x-h)
2
- = 1
2 2
Parabola/Quadratic Function
b a
2 2
y = ax + bx+ c f ( x)
= ax + bx+
c
Graph is a hyperbola that opens up and
hk , , vertices b
The graph is a parabola that opens up if
a > 0 or down if a < 0 and has a vertex
æ b æ b öö
at ç - , f ç - ÷
2a
2a
÷ .
è è øø
down, has a center at ( )
units up/down from the center and
asymptotes that pass through center with
b
slope ± .
a
Common Algebraic Errors
Error
Reason/Correct/Justification/Example
2
0
0 ¹ and 2 ¹ 2
Division by zero is undefined!
0
2
- 3 ¹ 9
( x
2 ) 3
¹ x
5
( ) 3
a a a
¹ +
b+
c b c
1
¹ x + x
2 3
x + x
-2 -3
a + bx
¹ 1+
bx
a
2
- 3 =- 9, (- 3) 2
= 9 Watch parenthesis!
x = xxx = x
a( x 1)
( )
- - ¹-ax- a
2 2 2 2 6
1 1 1 1
= ¹ + = 2
2 1+
1 1 1
A more complex version of the previous
error.
a+
bx a bx bx
= + = 1+
a a a a
Beware of incorrect canceling!
-a x- 1 =- ax+
a
Make sure you distribute the “-“!
( ) 2 2 2
x+ a ¹ x + a
( ) ( )( )
2 2
x + a ¹ x+
a
2 2 2
x+ a = x+ a x+ a = x + 2ax+
a
2 2 2 2
5= 25 = 3 + 4 ¹ 3 + 4 = 3+ 4 = 7
x+ a ¹ x+ a
See previous error.
+ ¹ + and n x a n x n More general versions of previous three
+ ¹ + a
errors.
( ) n n n
x a x a
( x+ ) ¹ ( x+
)
2 2
2 1 2 2
( 2x+ 2) ¹ 2( x+
1)
2 2
( ) ( )
2 2 2
2 x+ 1 = 2 x + 2x+ 1 = 2x + 4x+
2
( ) 2 2
2x+ 2 = 4x + 8x+
4
Square first then distribute!
See the previous example. You can not
factor out a constant if there is a power on
the parethesis!
2 2 2 2
- x + a ¹- x + a
( ) 1
- x 2 + a 2 = - x 2 + a
2 2
Now see the previous error.
æaö
a ab
¹
a
ç
1
÷
æaöæcö
ac
æb
ö c
=
è ø
=
ç
c
÷
b b
ç ÷ç ÷ =
æ ö æ ö è1øèbø
b
è ø
ç ÷ ç ÷
ècø ècø
æaö æa
ö
æaö
ç
ç ÷
b
÷ ç
b
÷
è ø æaöæ1ö
a
èb
ø ac
=
è ø
= ¹ c c
ç
b
֍
c
÷ =
æ ö è øè ø bc
c b
ç ÷
è1ø
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu.
© 2005 Paul Dawkins
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu.
© 2005 Paul Dawkins