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Basics of Quadratic Inequality or Inequations

Posted on - 23-02-2017

JEE Math QE

IIT JEE

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.

 
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