Multiple choice

Consider the equation: |x - 5|2 + 5 |x - 5| - 24 = 0 The sum of all the real roots of the above equation is:

  1. 2

  2. 3

  3. 8

  4. 10
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D Correct answer
Explanation

Let u = |x-5|. u^2 + 5u - 24 = 0. (u+8)(u-3) = 0. u = 3 or u = -8. Since u is an absolute value, u must be >= 0, so |x-5| = 3. x-5 = 3 or x-5 = -3. x = 8 or x = 2. Sum of roots = 8 + 2 = 10.

AI explanation

Assuming |x - 5| = y, the equation becomes a standard quadratic y^2 + 5y - 24 = 0. Factoring this yields (y + 8)(y - 3) = 0, so y is 3 since the absolute value must be positive. Substituting back gives |x - 5| = 3, which results in two equations: x - 5 = 3 and x - 5 = -3. Solving these yields the real roots 8 and 2, whose sum is 10.

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