<p>Let <em>α</em>, <em>β</em> be the roots of <em>x</em><sup>2</sup> + <em>bx</em> + 1 = 0. Then find the equation whose roots are <em>−</em>(<em>α</em> + 1/<em>β</em>) and <em>−</em>(<em>β</em> + 1/<em>α</em>).</p>
Step-by-Step Solution
Key Concept: Use Vieta's formulas (α + β = -b, αβ = 1) to express the new roots in terms of b, then construct the quadratic using sum and product of these transformed roots.
<p><strong>Step 1:</strong> From x² + bx + 1 = 0, by Vieta's formulas:</p><p>α + β = -b and αβ = 1</p><p><strong>Step 2:</strong> Since αβ = 1, we have 1/β = α and 1/α = β</p><p><strong>Step 3:</strong> The new roots are:</p><p>−(α + 1/β) = −(α + α) = −2α</p><p>−(β + 1/α) = −(β + β) = −2β</p><p><strong>Step 4:</strong> Sum of new roots = −2α − 2β = −2(α + β) = −2(−b) = 2b</p><p><strong>Step 5:</strong> Product of new roots = (−2α)(−2β) = 4αβ = 4(1) = 4</p><p><strong>Step 6:</strong> The required equation with sum 2b and product 4:</p><p>∴ Answer: <strong>x² − x(2b) + 4 = 0</strong></p>
Correct Answer: x^2 - x(2b) + 4 = 0