Multiple choice

Sum of the roots of the equation ${ \left| x-3 \right| }^{ 2 }+\left| x-3 \right| -2=0$ is

  1. $2$
  2. $4$
  3. $6$
  4. $16$
  5. $-2$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let u = |x-3|. u^2 + u - 6 = 0 => (u+3)(u-2) = 0. u = 2. |x-3| = 2 => x-3 = 2 or x-3 = -2. x = 5 or x = 1. Sum = 6.

AI explanation

Let u = |x - 3|, which transforms the equation into the quadratic u^2 + u - 2 = 0. Factoring this gives (u + 2)(u - 1) = 0, so the potential values for u are -2 and 1. Because u represents an absolute value, it must be non-negative, discarding -2 and leaving |x - 3| = 1. This gives x - 3 = 1 or x - 3 = -1, yielding x = 4 and x = 2. Adding these roots together gives 4 + 2 = 6.

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