Three Distinct Real Roots of Absolute Quadratic

CAT 2021 Slot 2 · QA · Hard · Quadratic Equations

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If rr is a constant such that ∣x2−4x−13∣=r|x^2 - 4x - 13| = r has exactly three distinct real roots, then the value of rr is

  1. A.

    15

  2. B.

    18

  3. C.

    17

  4. D.

    21

Answer

C

Explanation

Let g(x)=x2−4x−13=(x−2)2−17g(x) = x^2 - 4x - 13 = (x-2)^2 - 17. The vertex of the parabola g(x)g(x) is at (2,−17)(2, -17).

The graph of f(x)=∣x2−4x−13∣f(x) = |x^2 - 4x - 13| flips the negative portion of g(x)g(x) above the x-axis. The local maximum of f(x)f(x) occurs at the vertex x=2x = 2, where f(2)=∣−17∣=17f(2) = |-17| = 17.

For f(x)=rf(x) = r to have exactly three distinct roots, the horizontal line y=ry = r must touch the local maximum of the inverted parabola. Thus, r=17r = 17.

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