<p>The quadratic equation \(x^2 + bx + c = 0\) has distinct roots. If 2 is subtract from each root then result are the reciprocal of the original root. The value of \((b^2 + c^2)\) is:</p>
Step-by-Step Solution
Key Concept: If roots are α and β, then the condition (α-2) = 1/α and (β-2) = 1/β creates a system of equations relating the roots. Use Vieta's formulas to express b and c in terms of roots, then find b² + c².
<p><strong>Step 1:</strong> Let the distinct roots be α and β. Given condition: α - 2 = 1/α and β - 2 = 1/β</p><p><strong>Step 2:</strong> From α - 2 = 1/α: α² - 2α = 1 → α² - 2α - 1 = 0. Similarly β² - 2β - 1 = 0</p><p>Both roots satisfy the same equation x² - 2x - 1 = 0</p><p><strong>Step 3:</strong> Solve x² - 2x - 1 = 0: x = (2 ± √(4+4))/2 = (2 ± 2√2)/2 = 1 ± √2</p><p>So α = 1 + √2 and β = 1 - √2 (distinct roots ✓)</p><p><strong>Step 4:</strong> From Vieta's formulas: α + β = -b and αβ = c</p><p>α + β = (1 + √2) + (1 - √2) = 2, so b = -2</p><p>αβ = (1 + √2)(1 - √2) = 1 - 2 = -1, so c = -1</p><p><strong>Step 5:</strong> b² + c² = (-2)² + (-1)² = 4 + 1 = 5</p><p>∴ Answer: <strong>5</strong></p>
Correct Answer: D