CAT 2025

Slot 3

QA

Question & SolutionQues & Sol

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

1. Concept Used

• Topic: Algebra — Substitution, Quadratic Equations, and Vieta's Formulas
• Formula:

$$\text{Sum of roots of } ax^2 + bx + c = 0 \text{ is } -\dfrac{b}{a}$$

2. Calculation

Let $k = x^2 + 3x$. Then: $$f(x) = k(k + 2) = k^2 + 2k$$ $$f(x) + 1 = k^2 + 2k + 1 = (k+1)^2$$

So the equation becomes: $$\sqrt{(k+1)^2} = |k+1| = 9701$$

This gives two cases: $k + 1 = 9701$ or $k + 1 = -9701$.

Case 1: $k = 9700 \Rightarrow x^2 + 3x - 9700 = 0$

Discriminant $= 9 + 4(9700) = 38809 > 0$ — two real roots.

Sum of roots $= -3$.

Case 2: $k = -9702 \Rightarrow x^2 + 3x + 9702 = 0$

Discriminant $= 9 - 4(9702) = -38799 < 0$ — no real roots.

Only Case 1 yields real roots.

3. Solution

Answer = Option D ✅

The sum of all real roots is $\mathbf{-3}$.