<p>The sum of the roots of the equation, \(x^2 + |2x - 3| - 4 = 0\), is</p>
Step-by-Step Solution
Key Concept: The absolute value term creates two cases: when 2x - 3 ≥ 0 and when 2x - 3 < 0. You must solve both resulting quadratic equations separately and verify which roots satisfy their respective domain conditions.
<p><strong>Step 1:</strong> Split into cases based on the absolute value |2x - 3|.</p><p><strong>Case 1:</strong> When 2x - 3 ≥ 0, i.e., x ≥ 3/2:<br>x² + (2x - 3) - 4 = 0<br>x² + 2x - 7 = 0<br>x = (-2 ± √(4 + 28))/2 = (-2 ± √32)/2 = -1 ± 2√2</p><p>Check validity: x₁ = -1 + 2√2 ≈ 1.828 ≥ 3/2 ✓ (valid)<br>x₂ = -1 - 2√2 ≈ -3.828 < 3/2 ✗ (invalid)</p><p><strong>Case 2:</strong> When 2x - 3 < 0, i.e., x < 3/2:<br>x² - (2x - 3) - 4 = 0<br>x² - 2x - 1 = 0<br>x = (2 ± √(4 + 4))/2 = (2 ± 2√2)/2 = 1 ± √2</p><p>Check validity: x₃ = 1 + √2 ≈ 2.414 > 3/2 ✗ (invalid)<br>x₄ = 1 - √2 ≈ -0.414 < 3/2 ✓ (valid)</p><p><strong>Step 2:</strong> Sum of valid roots = (-1 + 2√2) + (1 - √2) = √2</p><p>∴ Answer: B</p>
Correct Answer: B