1.Algebra Booster

2.4 Algebra Booster
First we find the values of
Ê b ˆ
f( d), f( e), f Á-
.
Ë
˜
2a¯
Ï
Ê b ˆ¸
Maximum value = M = max Ìf( d), f( e),
f Á-
˜˝
Ó
Ë 2a¯˛
Ï
Ê b ˆ¸
Minimum value = m = min Ìf( d), f( e),
f Á-
˜˝
Ó
Ë 2a¯˛
Case VI: Maximum and Minimum values of a rational
function
2
ax + bx + c
Let y =
2
px + qx + r
To find the maximum and minimum values of y, we should
remember the following points:
(i) First we make it a quadratic equation of x.
(ii) Since x, y in R , then D ≥ 0
(iii) Then solve the quadratic equation of y.
(iv) If y Π[A, B], the maximum value = A and Minimum
Value = B
(v) If y Œ (– , A] » [B, ), the maximum and minimum
values are not defined.
(vi) If y Π(Р, ), the maximum and minimum values are
also not defined.
1.14 RESOLUTION OF A SECOND DEGREE
EXPRESSION IN X AND Y
Let f(x, y) = ax 2 + 2hxy + by 2 + 2gx + 2fy + c.
Now, f(x, y) = 0 gives
ax 2 + 2(hy + g)x + (by 2 + 2fy + c) = 0
fi
2 2
- 2( hy + g) ± 2 ( hy + g) - a( by + 2fy
+ c)
x =
2a
2 2
-( hy + g) ± ( hy + g) - a( by + 2 fy + c)
=
a
2 2 2
fi ax + hy + g =± ( h - ab) y + 2( hg - af ) y + ( g -ac)
Now in order that f(x, y) may be the product of two linear
factors of the form px + qy + r, the quantity under the radical
sign must be a perfect square, hence
(hg – af) 2 = (h 2 – ab)(g 2 – bc)
fi (h 2 g 2 + a 2 f 2 – 2afgh) = (g 2 h 2 – g 2 ab – h 2 bc + a 2 bc)
fi abc + 2fgh – af 2 – bg 2 – ch 2 = 0
a h g
fi h b f = 0
g f c
1.15 LOCATION OF THE ROOTS
Let f(x) = ax 2 + bx + c, where a π 0 and a and b be the roots
of f(x) = 0.
For simplicity, we can assume that a £ b.
b D
where a - -
- b+
D
= and b = , D = b 2 – 4ac
2a
2a
1. When both roots are +ve
0
(i) Sum of the roots > 0
(ii) Product of the roots > 0
(iii) For real roots, D ≥ 0.
2. When both roots are negative
Y
X¢
(i) Sum of the roots < 0.
(ii) Product of the roots > 0.
(iii) For real roots, D ≥ 0
3. When roots are of opposite signs
Y
X¢
Y
0
Y¢
(i) Product of the roots < 0
4. When both roots are
greater than k.
(i) D ≥ 0
(ii) af(k) > 0
(iii) a + b > 2k
5. When both roots are
less than k
(i) D ≥ 0
(ii) af(k) > 0
(iii) a + b < 2k
6. When k lies between
the roots
(i) D > 0
(ii) af(k) < 0
7. When both roots are
confined by k 1
and k 2
such that k 1
< k 2
(i) D ≥ 0
(ii) af(k 1
) > 0
0
X
X
X
k a
a
a
b
k
b
k
b
k 1 a b k 2