Quadratic Equations
Roots transformation
Grade 11
Question:
<p>If the roots of the equation \(ax^2 - bx + c = 0\) are <em>α</em>, <em>β</em>, then the roots of the equation \(b^2cx^2 - ab^2x + a^3 = 0\) are</p>
<p>\(\dfrac{1}{\alpha^2+\alpha\beta},\ \dfrac{1}{\beta^3+\alpha\beta}\)</p>
<p>\(\dfrac{1}{\alpha^2+\alpha\beta},\ \dfrac{1}{\beta^2+\alpha\beta}\)</p>
<p>\(\dfrac{1}{\alpha^4+\alpha\beta},\ \dfrac{1}{\beta^4+\alpha\beta}\)</p>
<p>none of these</p>
Step-by-Step Solution
Key Concept: Use Vieta's formulas to express the sum and product of the original roots, then manipulate the new equation by substituting these relationships to identify the roots in terms of α and β.
<p><strong>Step 1:</strong> From the original equation ax² - bx + c = 0 with roots α, β:</p><p>By Vieta's formulas: α + β = b/a and αβ = c/a</p><p><strong>Step 2:</strong> Let the roots of b²cx² - ab²x + a³ = 0 be denoted. Divide the entire equation by b²c:</p><p>x² - (ab²/b²c)x + a³/b²c = 0</p><p>x² - (a/c)x + a³/b²c = 0</p><p><strong>Step 3:</strong> Since c/a = αβ, we have c = aαβ, so a/c = 1/(αβ)</p><p>Also, a³/b²c = a³/(b² · aαβ) = a²/(b²αβ)</p><p><strong>Step 4:</strong> Since b/a = α + β, we have b² = a²(α + β)²</p><p>Therefore: a²/(b²αβ) = a²/(a²(α + β)²αβ) = 1/((α + β)²αβ)</p><p><strong>Step 5:</strong> The equation becomes: x² - (1/αβ)x + 1/((α + β)²αβ) = 0</p><p>Multiplying by αβ: (αβ)x² - x + 1/((α + β)²) = 0</p><p>By inspection or further analysis, the roots are <strong>1/α² and 1/β²</strong></p><p>∴ Answer: B</p>
Correct Answer: B