CAT 2025 Slot 3 QA Q11 Equations

Question 11 of the Quantitative Ability (QA) section, CAT 2025 Slot 3, with CAT's official answer and a worked solution.

If \( f ( x ) = \left( x ^ { 2 } + 3 x \right) \left( x ^ { 2 } + 3 x + 2 \right) \), then the sum of all real roots of the equation \( \sqrt { f ( x ) + 1 } = 9701 \), is

A
\( 6 \)
B
\( - 3 \)
C
\( - 6 \)
D
\( 3 \)

Answer: (B) - 3 Equations MCQ

Solution

Let \(u = x^2 + 3x\). Then \(f(x) + 1 = u(u + 2) + 1 = (u + 1)^2\).
So \(|u + 1| = 9701\): \(u = 9700\) or \(u = -9702\).
\(x^2 + 3x + 9702 = 0\) has negative discriminant; \(x^2 + 3x - 9700 = 0\) has two real roots with sum \(-3\).