Quadratic Equations
Roots transformation
Grade 11
Question:
<p>If <em>α</em>, <em>β</em> are the roots of \(ax^2 + bx + c = 0\) and \(\alpha+h\), \(\beta+h\) are the roots of \(px^2 + qx + r = 0\), then <em>h</em> =</p>
<p>\(-\dfrac{1}{2}\left(\dfrac{a}{b} - \dfrac{p}{q}\right)\)</p>
<p>\(\left(\dfrac{b}{a} - \dfrac{q}{p}\right)\)</p>
<p>\(\dfrac{1}{2}\left(\dfrac{b}{a} - \dfrac{q}{p}\right)\)</p>
<p>none of these</p>
Step-by-Step Solution
Key Concept: Use Vieta's formulas to relate the sum of roots in both equations. Since the roots are shifted by h, the difference in the sum of roots (divided by the ratio of leading coefficients) gives h directly.
<p><strong>Step 1:</strong> For equation ax² + bx + c = 0 with roots α, β:</p><p>α + β = -b/a (sum of roots)</p><p><strong>Step 2:</strong> For equation px² + qx + r = 0 with roots (α+h), (β+h):</p><p>(α + h) + (β + h) = -q/p</p><p>α + β + 2h = -q/p</p><p><strong>Step 3:</strong> Substitute α + β = -b/a into the equation from Step 2:</p><p>-b/a + 2h = -q/p</p><p><strong>Step 4:</strong> Solve for h:</p><p>2h = -q/p + b/a</p><p>2h = (bp - aq)/(ap)</p><p>∴ h = (bp - aq)/(2ap)</p>
Correct Answer: D