<p>Given the quadratic equation \(81x^2 + kx + 256 = 0\) and one real root is the cube of the other root. Find the value of \(k\).</p>
Step-by-Step Solution
Key Concept: If one root is the cube of the other, use Vieta's formulas: let roots be α and α³, then their product α·α³ = α⁴ equals the constant term ratio. This constraint determines α, which then determines k through the sum of roots.
<p><strong>Step 1:</strong> Let the roots be α and α³. By Vieta's formulas:</p><p>Product of roots: α · α³ = α⁴ = 256/81</p><p><strong>Step 2:</strong> Solve for α: α⁴ = 256/81 = (2⁴)/(3⁴)</p><p>Therefore: α² = ±16/9</p><p>Since α⁴ must be positive, we have α² = 16/9, giving α = ±4/3</p><p><strong>Step 3:</strong> Find k using the sum of roots:</p><p>Sum of roots: α + α³ = -k/81</p><p><strong>Case 1:</strong> If α = 4/3, then α³ = 64/27</p><p>Sum = 4/3 + 64/27 = 36/27 + 64/27 = 100/27</p><p>So: 100/27 = -k/81 → k = -300</p><p><strong>Case 2:</strong> If α = -4/3, then α³ = -64/27</p><p>Sum = -4/3 - 64/27 = -36/27 - 64/27 = -100/27</p><p>So: -100/27 = -k/81 → k = 300</p><p><strong>Verification:</strong> With k = -300: 81x² - 300x + 256 = 0 has roots 4/3 and 64/27 ✓</p><p>∴ Answer: k = -300 or k = 300 (if only one answer required, typically k = <strong>-300</strong>)</p>
Correct Answer: -300