<p>A value of <i>b</i> for which the equations \(x^2 + bx - 1 = 0\) and \(x^2 + x + b = 0\) have one root in common is</p>
Step-by-Step Solution
Key Concept: If two quadratic equations share a common root α, then α satisfies both equations simultaneously. Subtracting one equation from the other eliminates the x² term and gives a linear equation that the common root must satisfy.
<p><strong>Step 1:</strong> Let α be the common root of both equations.</p><p>Then: α² + bα - 1 = 0 ... (1)</p><p>And: α² + α + b = 0 ... (2)</p><p><strong>Step 2:</strong> Subtract equation (2) from equation (1):</p><p>(α² + bα - 1) - (α² + α + b) = 0</p><p>bα - 1 - α - b = 0</p><p>α(b - 1) = b + 1</p><p>α = (b + 1)/(b - 1), provided b ≠ 1</p><p><strong>Step 3:</strong> Substitute this expression for α back into equation (2):</p><p>[(b + 1)/(b - 1)]² + (b + 1)/(b - 1) + b = 0</p><p><strong>Step 4:</strong> Multiply through by (b - 1)²:</p><p>(b + 1)² + (b + 1)(b - 1) + b(b - 1)² = 0</p><p>(b² + 2b + 1) + (b² - 1) + b(b² - 2b + 1) = 0</p><p>b² + 2b + 1 + b² - 1 + b³ - 2b² + b = 0</p><p>b³ + 3b = 0</p><p>b(b² + 3) = 0</p><p><strong>Step 5:</strong> Since b² + 3 > 0 for all real b, we get b = 0.</p><p>∴ Answer: B</p>
Correct Answer: B