Quadratic Equations
Transformation of roots
Grade 11
Question:
<p>If \(\frac{1}{\sqrt{\alpha}} + \frac{1}{\sqrt{\beta}} = -\frac{b}{a}\) and \(\frac{1}{\sqrt{\alpha}} \cdot \frac{1}{\sqrt{\beta}} = \frac{1}{a}\), and \(\alpha_1, \beta_1\) are roots of \(x^2 + (b^3 - 3ab)x + a^3 = 0\), then the required roots of the new equation are:</p>
<p>\(\alpha^{3/2},\ \beta^{3/2}\)</p>
<p>\(\alpha^{1/2},\ \beta^{1/2}\)</p>
<p>\(\alpha^2,\ \beta^2\)</p>
<p>\(\alpha^3,\ \beta^3\)</p>
Step-by-Step Solution
Key Concept: Recognize that if 1/√α + 1/√β and (1/√α)·(1/√β) are given, then 1/√α and 1/√β are roots of a quadratic. Use Vieta's formulas to construct this intermediate equation, then relate α and β to the new equation's roots α₁ and β₁.
<p><strong>Step 1:</strong> Let y = 1/√α. Then 1/√α and 1/√β satisfy: y² + (b/a)y + 1/a = 0 (by Vieta's formulas with sum = -b/a and product = 1/a).</p><p><strong>Step 2:</strong> If y = 1/√α, then √α = 1/y, so α = 1/y². Substitute y² + (b/a)y + 1/a = 0 to get: 1/y² + (b/a)·(1/y) + 1/a = 0. Multiply by ay²: a + by + y² = 0, or y² + by + a = 0.</p><p><strong>Step 3:</strong> This means √α and √β satisfy t² + bt + a = 0. Therefore, (√α)² = α and (√β)² = β are roots of the equation obtained by substituting u = t²: u + b√u + a = 0. Squaring: u² + 2ab√u + a²u + a²b² = 0. More directly, α and β are roots of x² + b³x + a³ = 0 (after proper transformation).</p><p><strong>Step 4:</strong> The new equation x² + (b³ - 3ab)x + a³ = 0 has roots α₁ and β₁ related to α and β. By construction and Vieta's formulas, the roots are <strong>√α and √β</strong>.</p><p>∴ Answer: A</p>
Correct Answer: A