Question

If the sum of the roots of the quadratic equation is $3$ and sum of their cubes is $63$, then the quadratic equation is $x^2-3x-m=0$. The value of $m$ is

Answer 1

Let the roots be $a$ and $b$
$a^3+b^3=63$
$\Rightarrow a+b=3$
Cubing both sides,
$(a+b{)}^3=27$
$a^3+b^3+3ab(a+b)=27$
$63+3ab\left(3\right)=27$ $($putting the values of $a^3+b^3$ and $a+b)$
$9ab=27-63$
$9ab=-36$
$ab=-4$
The general equation is $x^2-Sx+P=0$ will be $x^2-3x-4=0$