Question

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

Answer 1

Let the two roots be $\alpha$ and $\beta$

$\alpha +\beta =5$ , ${\alpha }^3+{\beta }^3=35$

$(\alpha +\beta {)}^3={\alpha }^3+{\beta }^3+3\alpha \beta (\alpha +\beta )$

$(5{)}^3=35+3\alpha \beta (5)$

$125=35+15\alpha \beta$

$90=15\alpha \beta$

$\therefore \frac{90}{15}=\alpha \beta$

$\therefore \alpha \beta =6$

Quadratic $E{q}^n$ :

$x^2-\left(sumofroots\right)x+productofroots=0$

$x^2-(\alpha +\beta )x+\alpha \beta =0$

$x^2-5x+6=0$