Theory of EquationseasyFree

Theory of Equations: Quadratic Polynomial Positive Every Real Greatest Absolute V

JEE Maths question with a full step-by-step solution.

Question

If the quadratic polynomial

P(x) = 4x^2 + (k+8)x + 9

k \in \mathbb{I}

, is positive for every real

x

, then the greatest absolute value of

k

20

19

correct

4

3

Solution

Step 1: For

P(x) > 0

for all

x \in \mathbb{R}

, the discriminant must be negative:

D = (k+8)^2 - 4 \cdot 4 \cdot 9 < 0 \implies (k+8)^2 - 144 < 0 \implies (k-4)(k+20) < 0

-20 < k < 4

Step 2: Integer values of

k

k \in \{-19, -18, \ldots, 3\}

. Step 3: Greatest absolute value:

\max\{|-19|, |3|\} = 19

. Correct answer: (2)

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