Chapter: Quadratic Equations· Mathematics Part 1 (Algebra)
Answer: \(x^2 - 7x + 12 = 0\)
Let the roots be \(\alpha\) and \(\beta\). \(\alpha + \beta = 7\), \(\alpha^3 + \beta^3 = 91\).
\(\alpha^3 + \beta^3 = (\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta)\)
\(91 = 7^3 - 3\alpha\beta(7)\)
\(91 = 343 - 21\alpha\beta\)
\(21\alpha\beta = 252 \Rightarrow \alpha\beta = 12\)
Required equation: \(x^2 - (\alpha + \beta)x + \alpha\beta = 0\)
\(\therefore x^2 - 7x + 12 = 0\)
(Its roots are 3 and 4: \(27 + 64 = 91\) ✓.)