<p>If \((1 - p)\) is a root of quadratic equation \(x^2 + px + (1 - p) = 0\), then find its roots.</p>
Step-by-Step Solution
Key Concept: Substitute the given root (1-p) into the equation to find p, then solve the resulting quadratic. This transforms an unknown into a constraint that determines the equation completely.
<p><strong>Step 1:</strong> Since (1-p) is a root, substitute x = (1-p) into the equation:</p><p>(1-p)² + p(1-p) + (1-p) = 0</p><p><strong>Step 2:</strong> Expand (1-p)²:</p><p>1 - 2p + p² + p - p² + 1 - p = 0</p><p><strong>Step 3:</strong> Simplify by combining like terms:</p><p>1 - 2p + p² + p - p² + 1 - p = 0</p><p>2 - 2p = 0</p><p>p = 1</p><p><strong>Step 4:</strong> Substitute p = 1 back into the original equation:</p><p>x² + x + (1-1) = 0</p><p>x² + x = 0</p><p>x(x + 1) = 0</p><p><strong>Step 5:</strong> Factor to find roots:</p><p>x = 0 or x = -1</p><p><strong>Verification:</strong> When p = 1, (1-p) = 0, which is indeed one of our roots. ✓</p><p>∴ <strong>Roots are: x = 0, -1</strong></p>
Correct Answer: x = 0, -1