Step-by-Step Solution
Key Concept: A product equals zero if and only if at least one factor equals zero. You must solve each factor separately, then check which solutions satisfy the domain restriction from the square root (radicand ≥ 0).
<p><strong>Step 1: Set each factor to zero</strong></p><p>√(x-2) = 0 OR x² - 4x - 5 = 0</p><p><strong>Step 2: Solve √(x-2) = 0</strong></p><p>Squaring both sides: x - 2 = 0 → x = 2</p><p><strong>Step 3: Solve x² - 4x - 5 = 0</strong></p><p>Factoring: (x - 5)(x + 1) = 0 → x = 5 or x = -1</p><p><strong>Step 4: Apply domain restriction from √(x-2)</strong></p><p>We need x - 2 ≥ 0 → x ≥ 2</p><p>Check each solution:</p><p>• x = 2: 2 ≥ 2 ✓ Valid</p><p>• x = 5: 5 ≥ 2 ✓ Valid</p><p>• x = -1: -1 ≥ 2 ✗ Invalid</p><p><strong>∴ Answer: x = 2 or x = 5</strong></p>
Correct Answer: x = 2 or x = 5