Question

If $\alpha +\beta =5$ and ${\alpha }^3+{\beta }^3=35$, find the quadratic equation whose roots are $\alpha$ and $\beta$.

Answer 1

We know,

$(a+b{)}^3=a^3+3a^{2}b+3a{b}^2+b^3$

$⟹a^3+b^3=(a+b{)}^3-3ab(a+b)$

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

$35-125=$$-3$$\alpha$$\beta$$(\alpha +\beta )$

$\alpha$$\beta$$(\alpha +\beta )$$=30$

$\alpha$$\beta$$=6$

Sum of roots $=5$

Product of roots$=6$

The quadratic equation is $x^2-\left(Sumofroots\right)x+Productofroot=0$

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