Quadratic Equations
Equations with Transformed Roots
Grade None
Question:
<p>If <i>α</i> and <i>β</i> are the roots of the equation <i>2x</i><sup>2</sup> + 3<i>x</i> + 4 = 0, then the equation whose roots are <i>α</i><sup>2</sup> and <i>β</i><sup>2</sup>, is</p>
<p>(a) \(4x^2 + 7x + 16 = 0\)</p>
<p>(b) \(4x^2 + 7x + 6 = 0\)</p>
<p>(c) \(4x^2 + 7x + 1 = 0\)</p>
<p>(d) \(4x^2 - 7x + 16 = 0\)</p>
Step-by-Step Solution
Key Concept: Use Vieta's formulas to find α + β and αβ from the original equation, then construct a new equation with roots α² and β² by finding their sum (α² + β²) and product (α²β²).
<p><strong>Step 1: Apply Vieta's formulas to 2x² + 3x + 4 = 0</strong></p><p>For the equation 2x² + 3x + 4 = 0 with roots α and β:</p><p>α + β = -3/2</p><p>αβ = 4/2 = 2</p><p><strong>Step 2: Find α² + β² (sum of new roots)</strong></p><p>α² + β² = (α + β)² - 2αβ</p><p>α² + β² = (-3/2)² - 2(2)</p><p>α² + β² = 9/4 - 4 = 9/4 - 16/4 = -7/4</p><p><strong>Step 3: Find α²β² (product of new roots)</strong></p><p>α²β² = (αβ)² = (2)² = 4</p><p><strong>Step 4: Form the new equation with roots α² and β²</strong></p><p>If the new equation is Ax² + Bx + C = 0, then:</p><p>Sum of roots = -B/A and Product of roots = C/A</p><p>Let A = 4: -B/4 = -7/4, so B = 7</p><p>C/4 = 4, so C = 16</p><p><strong>Step 5: Verify the equation</strong></p><p>The equation is 4x² + 7x + 16 = 0</p><p>∴ Answer: A</p>
Correct Answer: A