Multiple choice

What are the roots of the equation |x2 - x - 6| = x + 2?

  1. -2, 1, 4

  2. 0, 2, 4

  3. 0, 1, 4

  4. -2, 2, 4
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D Correct answer
Explanation

Solve |x^2 - x - 6| = x + 2. Case 1: x^2 - x - 6 = x + 2 => x^2 - 2x - 8 = 0 => (x-4)(x+2) = 0, so x=4, -2. Case 2: x^2 - x - 6 = -(x + 2) => x^2 - 4 = 0 => x = 2, -2. Combining these, the roots are -2, 2, 4.

AI explanation

To solve the absolute value equation |x^2 - x - 6| = x + 2, split it into two cases based on the sign of the quadratic expression. In the first case, assuming x^2 - x - 6 is positive or zero, the equation becomes x^2 - x - 6 = x + 2, which rearranges to x^2 - 2x - 8 = 0. Factoring gives (x - 4)(x + 2) = 0, yielding potential roots x = 4 and x = -2, and both satisfy the original equation. In the second case, assuming the quadratic is negative, the equation becomes -(x^2 - x - 6) = x + 2, which rearranges to x^2 - 4 = 0. This yields potential roots x = 2 and x = -2, and substituting x = 2 into the original equation confirms it is valid. The full set of roots is -2, 2, 4.

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