<p>Sum of solutions of the equation \(|x|^3 - 4|x|^2 + 3|x| = 0\) is</p>
Step-by-Step Solution
Key Concept: Factor out |x| to get |x|(|x|² - 4|x| + 3) = 0, then recognize that for each positive solution a, we have both x = a and x = -a as solutions, making the sum zero by symmetry.
<p><strong>Step 1:</strong> Factor the equation: |x|³ - 4|x|² + 3|x| = 0</p><p>|x|(|x|² - 4|x| + 3) = 0</p><p><strong>Step 2:</strong> Factor the quadratic: |x|(|x| - 1)(|x| - 3) = 0</p><p>This gives |x| = 0, |x| = 1, or |x| = 3</p><p><strong>Step 3:</strong> Convert each to x solutions:</p><p>• |x| = 0 ⟹ x = 0</p><p>• |x| = 1 ⟹ x = ±1</p><p>• |x| = 3 ⟹ x = ±3</p><p><strong>Step 4:</strong> Sum all solutions: 0 + 1 + (-1) + 3 + (-3) = 0</p><p>∴ Answer: C (sum = 0)</p>
Correct Answer: C