Question

Form a quadratic equation whose sum of roots is $3$ and sum of their cubes is $63$.

Answer 1

Quad equation =

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

$\alpha +\beta =3$

${\alpha }^3+{\beta }^3=63$

$63=(\alpha +\beta {)}^3-3\alpha \beta (\alpha +\beta )$

$63=(3{)}^3-3\alpha \beta (3)$

$63=27-9\alpha \beta$

$9\alpha \beta =-36$

$\alpha \beta =-4$

equation is $x^2-\left(3\right)x+(-4)$

$=x^2-3x-4=0$