Question

The sum of two roots of a quadratic equation is 5 and sum of their cubes is 35, find the equation.

Answer 1

Let the roots be $\alpha \mathrm{and}\beta$.
Sum of roots = 5
$\alpha +\beta =5$
Also,
$$\begin{gathered} {\alpha }^3+{\beta }^3=35 \\ \Rightarrow {\alpha }^3+{\beta }^3={\left(\alpha +\beta \right)}^3-3\alpha \beta \left(\alpha +\beta \right) \\ \Rightarrow 35=5^3-3\alpha \beta \left(5\right) \\ \Rightarrow 15\alpha \beta =125-35 \\ \Rightarrow \alpha \beta =6 \end{gathered}$$
The general quadratic equation is
$x^2-\left(\alpha +\beta \right)x+\alpha \beta =0$
So, the required quadratic equation will be
$$\begin{gathered} x^2-\left(5\right)x+6=0 \\ \Rightarrow x^2-5x+6=0 \end{gathered}$$