Quadratic Equations
Roots and Transformation
Grade 11
Question:
<p>If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2 - mx + 2 = 0\) and \(\alpha + \dfrac{1}{\beta}\), \(\beta + \dfrac{1}{\alpha}\) are the roots of the equation \(x^2 - px + q = 0\), then the value of \(2q\) equals:</p>
<p>1</p>
<p>3</p>
<p>6</p>
<p>9</p>
Step-by-Step Solution
Key Concept: Use Vieta's formulas on the original equation to express α + β and αβ, then find the sum and product of the new roots in terms of m, which directly gives you the coefficient q.
<p><strong>Step 1:</strong> From <em>x</em>² − <em>mx</em> + 2 = 0, by Vieta's formulas:</p><p>α + β = <em>m</em> and αβ = 2</p><p><strong>Step 2:</strong> Find the sum of new roots:</p><p>(α + 1/β) + (β + 1/α) = α + β + 1/β + 1/α = α + β + (α + β)/(αβ)</p><p>= <em>m</em> + <em>m</em>/2 = 3<em>m</em>/2 = <em>p</em></p><p><strong>Step 3:</strong> Find the product of new roots:</p><p>(α + 1/β)(β + 1/α) = αβ + α·(1/α) + (1/β)·β + (1/β)·(1/α)</p><p>= αβ + 1 + 1 + 1/(αβ) = 2 + 2 + 1/2 = 9/2 = <em>q</em></p><p><strong>Step 4:</strong> Calculate 2<em>q</em>:</p><p>2<em>q</em> = 2 × 9/2 = 9</p><p>∴ Answer: D (which is 9)</p>
Correct Answer: D