If $\alpha ,\beta$ are the roots of the quadratic
equation $a{x}^2+bx+c=0$ then the quadratic equation whose roots are ${\alpha }^3,{\beta }^3$ is

Put $y={\alpha }^3\Rightarrow \alpha =y^{1/3}$

As $\alpha$ is root of $a{x}^2+bx+c=0$, we get

$a{y}^{2/3}+b{y}^{1/3}+c=0\Rightarrow y^{1/3}\left(a{y}^{1/3}+b\right)=-c$

Cubing both the side, we get

$\begin{array}{l}y{\left(a{y}^{1/3}+b\right)}^3=-c^3\\ \Rightarrow y\left[a^{3}y+b^3+3ab{y}^{1/3}\left(a{y}^{1/3}+b\right)\right]=-c^3\end{array}$

$\Rightarrow y\left[a^{3}y+b^3+3ab(-c)\right]=-c^3$

$\Rightarrow a^3{y}^2+\left(b^3-3abc\right)y+c^3=0$