<p>If <em>α</em>, <em>β</em> are the roots of \(x^2 + px + q = 0\) and <em>γ</em>, <em>δ</em> are the roots of \(x^2 + px + r = 0\), then \(\dfrac{(\alpha-\gamma)(\alpha-\delta)}{(\beta-\gamma)(\beta-\delta)} =\)</p>
Step-by-Step Solution
Key Concept: Use Vieta's formulas to express the roots implicitly, then evaluate each quadratic at the opposite root to create a ratio of function values that simplifies elegantly.
<p><strong>Step 1:</strong> Since γ, δ are roots of x² + px + r = 0, we can write:</p><p>(x - γ)(x - δ) = x² + px + r</p><p><strong>Step 2:</strong> Evaluate this quadratic at x = α:</p><p>(α - γ)(α - δ) = α² + pα + r</p><p><strong>Step 3:</strong> Since α is a root of x² + px + q = 0, we have α² + pα + q = 0, so α² + pα = -q</p><p>Therefore: (α - γ)(α - δ) = -q + r = r - q</p><p><strong>Step 4:</strong> Similarly, evaluate the quadratic at x = β:</p><p>(β - γ)(β - δ) = β² + pβ + r = -q + r = r - q</p><p><strong>Step 5:</strong> The ratio becomes:</p><p>$$\frac{(\alpha-\gamma)(\alpha-\delta)}{(\beta-\gamma)(\beta-\delta)} = \frac{r-q}{r-q} = 1$$</p><p>∴ Answer: <strong>A (which equals 1)</strong></p>
Correct Answer: A