<p>The given equation is \(|x - 2|^2 + |x - 2| - 2 = 0\). Solve for \(x\) when \(x \geq 2\).</p>
Step-by-Step Solution
Key Concept: Convert the absolute value equation to a quadratic by substitution. Check validity of solutions against the domain restriction.
<p><strong>Case I:</strong> If $x \geq 2$, then $(x - 2)^2 + (x - 2) - 2 = 0$</p><p><strong>Step 1:</strong> Let $u = x - 2$, so $u^2 + u - 2 = 0$</p><p><strong>Step 2:</strong> Factoring: $(u + 2)(u - 1) = 0$</p><p><strong>Step 3:</strong> So $u = -2$ or $u = 1$</p><p><strong>Step 4:</strong> Thus $x - 2 = 1$ gives $x = 3$ (valid for $x \geq 2$)</p><p>The solution for $x \geq 2$ is $x = 3$.</p>
Correct Answer: \(3\) or \(0\)