Question

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

• A. $4$
• B. $-4$
• C. $2$
• D. $-2$

Hint:

When dealing with modulus, use substitution to reduce the equation and consider domain constraints.

Answer 1

The Correct Option is A

Let $y = |x - 2|$, then the equation becomes $y^2 + y - 2 = 0$
Solve: $y = \frac{-1 \pm \sqrt{1 + 8}}{2} = \frac{-1 \pm 3}{2} \Rightarrow y = 1, -2$
Only $y = 1$ is valid (since $|x - 2| \geq 0$)
So, $|x - 2| = 1 \Rightarrow x = 1$ or $x = 3$
Sum of real roots = $1 + 3 = 4$