<p>If the sum of the roots of an equation is 2 and the sum of their cubes is 98, then find the equation.</p>
Step-by-Step Solution
Key Concept: Use Vieta's formulas (sum and product of roots) combined with the identity a³ + b³ = (a+b)³ - 3ab(a+b) to find the product of roots, then construct the quadratic equation.
<p><strong>Step 1:</strong> Let the roots be α and β. Given: α + β = 2 and α³ + β³ = 98</p><p><strong>Step 2:</strong> Use the identity α³ + β³ = (α + β)³ - 3αβ(α + β)</p><p>Substitute: 98 = (2)³ - 3αβ(2)</p><p>98 = 8 - 6αβ</p><p>6αβ = 8 - 98 = -90</p><p>αβ = -15</p><p><strong>Step 3:</strong> A quadratic equation with roots α and β is: x² - (α + β)x + αβ = 0</p><p>x² - (2)x + (-15) = 0</p><p><strong>∴ Answer: x² - 2x - 15 = 0</strong></p>
Correct Answer: x^2 - 2x - 15 = 0