Multiple choice

The sum of the real roots of the equation $|x - 2|^2 +|x - 2| -2 = 0$.

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

Let u = |x-2|. Equation: u^2 + u - 2 = 0. (u+2)(u-1) = 0. u = 1 or u = -2. Since u is absolute value, u = 1. |x-2| = 1. x-2 = 1 or x-2 = -1. x = 3 or x = 1. Sum of roots = 3 + 1 = 4.

AI explanation

To solve the equation, we use the substitution method by letting y equal the absolute value of (x minus 2). The equation transforms into the quadratic equation y squared plus y minus 2 equals 0. We factor this quadratic equation to get (y plus 2) times (y minus 1) equals 0, which gives the roots y equals negative 2 and y equals 1. Since y represents an absolute value, it must be non-negative, so we discard negative 2. We then set the absolute value of (x minus 2) equal to 1, which gives two linear equations: x minus 2 equals 1 and x minus 2 equals negative 1. Solving these gives the real roots x equals 3 and x equals 1. The sum of these real roots is 3 plus 1, which equals 4.

Keep practicing — related questions