Quadratic Inequations
Let f(x) = ax2
+ bx + c be a quadratic expression. Then inequations of the type f(x) £
0 or f(x) ≥ 0 are known as quadratic
inequations. The study of these can be easily done by taking the corresponding
quadratic expression and by applying the basic results of quadratic expression.
Example 1
Find the value of
‘a’ for which ax2 + (a - 3) x + 1 < 0 for at least one positive
real x .
Solution:
Let f(x) = ax2 + (a - 3)x
+ 1
Case(i):
If a > 0 , then f(x) will be
negative only for those values of x which lie between the roots. From the
graphs we can see that f(x) will be less than zero for at least one positive
real x, when f(x) = 0 has distinct roots and at least one of these roots is
positive real root. .
Case
(ii):
If a
< 0, then f(x) will be positive only for those values of x which lie between
the roots. .
As the interval between
the roots can not cover all the positive real numbers implies f(x) < 0 for
at least one positive real x
For this D > 0, i.e. (a - 3)2
- 4a > 0 .
&⇒
a < 1 or a > 9. (1).
Both the roots are non-positive
&⇒
sum £ 0 and product ≥ 0
&⇒
a ≥ 3 and (1) is satisfied
&⇒
at least one root is positive if a < 3, and (1) is satisfied …(2)
Combining (1) and (2), we get a <
1 so that, 0 < a < 1.
Case (iii)
If a = 0, f(x) = -3x + 1
&⇒
f(x) < 0 " x > 1/3
Hence the required set of values of
‘a’ is (-¥, 1)
Example 2
Find
the values of ‘a’ for which 4t - (a - 4) 2t +
< 0 " t ∈ (1, 2) .
Solution:
Let 2t = x
and f(x) = x2 - (a - 4)x + 
a
We want f(x) < 0 "
x ∈ ( 21, 22)
i.e. "
x ∈ (2, 4).
(i) Since
coefficient of x2 in f(x) is positive, f(x) < 0 for some x only
when roots of f(x) = 0 are real and distinct
&⇒
D > 0
&⇒
a2 - 17a + 16 > 0 are a > 16 …. (1).
(ii) Since we want
f(x) < 0 " x ∈ (2, 4), one
of the roots of
f(x) should be less than 2 and the other must be greater than 4
i.e. f(2) < 0 and f(4) < 0 .
a < - 48 and a
> 128/7, which is not possible .
Hence no
such ‘a’ exist.
