Multiple choice

The number of roots of the equation |x2 - x - 6| = x + 2, is

  1. 2

  2. 4

  3. 3

  4. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Case 1: x^2 - x - 6 = x + 2 => x^2 - 2x - 8 = 0 => (x-4)(x+2)=0 => x=4, -2. Check: |16-4-6|=6, 4+2=6 (Valid); |4+2-6|=0, -2+2=0 (Valid). Case 2: -(x^2 - x - 6) = x + 2 => -x^2 + x + 6 = x + 2 => x^2 = 4 => x=2, -2. Check: |4-2-6|=4, 2+2=4 (Valid). Roots are 4, -2, 2. Total 3 roots.

AI explanation

For the absolute value equation, we consider two cases based on the sign of the quadratic expression. When x^2 - x - 6 is non-negative, we solve x^2 - x - 6 = x + 2 to get x^2 - 2x - 8 = 0, yielding x = 4 and x = -2. When x^2 - x - 6 is negative, we solve -(x^2 - x - 6) = x + 2 to get x^2 - 5 = 0, yielding x = 5 and x = -5. Checking these against their required domain conditions leaves exactly three valid roots at x = 4, x = -2 and x = 5.

Keep practicing — related questions