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:

For equations involving modulus, substitute \[ |x-a|=y \] with \(y\geq0\). This converts the equation into a simpler algebraic equation.

Answer 1

The Correct Option is A

Step 1: Substitute to simplify.
Let $y=|x-2|$, so $y\ge 0$. The equation $|x-2|^2+|x-2|-2=0$ becomes $y^2+y-2=0$.
Step 2: Factor the quadratic.
$y^2+y-2=(y+2)(y-1)=0$, so $y=-2$ or $y=1$.
Step 3: Reject the invalid root.
Since $y=|x-2|\ge 0$, the value $y=-2$ is impossible; keep $y=1$.
Step 4: Solve the modulus.
$|x-2|=1$ gives $x-2=1$ or $x-2=-1$.
Step 5: Find the roots.
So $x=3$ or $x=1$.
Step 6: Add the real roots.
Their sum is $3+1=4$.
\[ \boxed{4} \]