Multiple choice

The sum of all real roots of the equation ${ \left| x-2 \right| }^{ 2 }+\left| x-2 \right| -2=0$ is

  1. $7$
  2. $4$
  3. $1$
  4. $5$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let y = |x-2|. y^2 + y - 2 = 0. (y+2)(y-1) = 0. Since y >= 0, y = 1. |x-2| = 1, so x-2 = 1 or x-2 = -1. x = 3 or x = 1. Sum = 4.

AI explanation

Let u = |x-2|. The equation becomes u^2 + u - 2 = 0. Factoring this quadratic gives (u - 1)(u + 2) = 0, so u = 1 or u = -2. Since u represents an absolute value, it cannot be negative, leaving only u = 1. Substituting back gives |x-2| = 1, which means x-2 = 1 or x-2 = -1. The two real roots are x = 3 and x = 1, and their sum is 4.

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