Multiple choice

What will be the sum of the squares of the distinct roots of the equation |2x2 - 3x - 5| = x + 1?

  1. 12

  2. 14

  3. 16

  4. 18
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Solve |2x^2 - 3x - 5| = x + 1. Case 1: 2x^2 - 3x - 5 = x + 1 => 2x^2 - 4x - 6 = 0 => x^2 - 2x - 3 = 0 => (x-3)(x+1) = 0. Roots: 3, -1. Case 2: 2x^2 - 3x - 5 = -(x + 1) => 2x^2 - 2x - 4 = 0 => x^2 - x - 2 = 0 => (x-2)(x+1) = 0. Roots: 2, -1. Distinct roots are 3, -1, 2. Sum of squares = 3^2 + (-1)^2 + 2^2 = 9 + 1 + 4 = 14.

AI explanation

The equation |2x2 - 3x - 5| = x + 1 requires x + 1 to be non-negative, meaning x must be greater than or equal to -1. Splitting into cases, we first solve 2x2 - 3x - 5 = x + 1, which simplifies to 2x2 - 4x - 6 = 0 or (x - 3)(x + 1) = 0, giving valid roots 3 and -1. For the second case, 2x2 - 3x - 5 = -x - 1 simplifies to 2x2 - 2x - 4 = 0 or (x - 2)(x + 1) = 0, giving valid roots 2 and -1. Since the distinct roots are 3, 2, and -1, their squares are 9, 4, and 1, which sum to 14.

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