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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)=4x2+(k+8)x+9P(x) = 4x^2 + (k+8)x + 9, k∈Ik \in \mathbb{I}, is positive for every real xx, then the greatest absolute value of kk is
A2020
B1919correct
C44
D33
Solution
Step 1: For P(x)>0P(x) > 0 for all x∈Rx \in \mathbb{R}, the discriminant must be negative:
D=(k+8)2−4⋅4⋅9<0  ⟹  (k+8)2−144<0  ⟹  (k−4)(k+20)<0D = (k+8)^2 - 4 \cdot 4 \cdot 9 < 0 \implies (k+8)^2 - 144 < 0 \implies (k-4)(k+20) < 0
−20<k<4-20 < k < 4
 Step 2: Integer values of kk: k∈{−19,−18,…,3}k \in \{-19, -18, \ldots, 3\}. Step 3: Greatest absolute value: max⁡{∣−19∣,∣3∣}=19\max\{|-19|, |3|\} = 19. Correct answer: (2)
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