<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 α and β are roots of x² + bx + c = 0, then α - 2 = 1/α and β - 2 = 1/β. Use Vieta's formulas combined with these transformation equations to find relationships between b and c.
<p><strong>Step 1:</strong> Let α and β be the distinct roots of x² + bx + c = 0.</p><p>By Vieta's formulas: α + β = -b and αβ = c</p><p><strong>Step 2:</strong> Given condition: (α - 2) = 1/α and (β - 2) = 1/β</p><p>From α - 2 = 1/α: α² - 2α = 1 ⟹ α² - 2α - 1 = 0</p><p>From β - 2 = 1/β: β² - 2β = 1 ⟹ β² - 2β - 1 = 0</p><p><strong>Step 3:</strong> Both α and β satisfy t² - 2t - 1 = 0</p><p>Therefore: x² + bx + c = x² - 2x - 1</p><p>Comparing: b = -2 and c = -1</p><p><strong>Step 4:</strong> Verify roots are distinct: Δ = 4 + 4 = 8 > 0 ✓</p><p>Roots are 1 ± √2 (distinct)</p><p><strong>Step 5:</strong> Calculate b² + c² = (-2)² + (-1)² = 4 + 1 = 5</p><p>∴ Answer: <strong>5</strong></p>
Correct Answer: D