<p>The number of real solutions of the equation \(|x^2 - 3|x| + 2| = 0\) is:</p>
Step-by-Step Solution
Key Concept: |expression| = 0 iff expression = 0. So x^2 - 3|x| + 2 = 0. Let t = |x| \geq 0: t^2 - 3t + 2 = 0 \to t = 1 or t = 2. Each gives two x values (\pm1 and \pm2). Total = 4 solutions.
Notice that the best first move is to reveal the hidden structure in the expression. A clever move here is to rewrite the problem in the form where the standard theorem or identity applies cleanly. \(|x^2-3|x|+2|=0 \Rightarrow x^2-3|x|+2=0\). Let \(t=|x|\geq0\): \(t^2-3t+2=0\Rightarrow(t-1)(t-2)=0\Rightarrow t=1\) or \(t=2\). \(|x|=1\Rightarrow x=\pm1\); \(|x|=2\Rightarrow x=\pm2\). Total: 4 real solutions. Now, we invoke the power of that idea, simplify patiently, and then check that the final answer really fits the original problem.
Correct Answer: 1