Algebra question: RMO/INMO problem-solving practice

Question:

If $\alpha$ , $\beta$ , $\gamma$ be the roots of the cubic equation $ax^{3}+3bx^{2}+3cx+d=0$ . Prove that the equation in y whose roots are $\frac{\beta\gamma-\alpha^{2}}{\beta+\gamma-2\alpha} + \frac{\gamma\alpha-\beta^{2}}{\gamma+\\alpha-2\beta} + \frac{\alpha\beta-\gamma^{2}}{\alpha+\beta-2\gamma}$ is obtained by the transformation $axy+b(x+y)+c=0$ . Hence, form the equation with above roots.

Solution:

Given that $\alpha$ , $\beta$ , $\gamma$ are the roots of the equation:

$ax^{3}+3bx^{2}+3cx+d=0$ …call this equation I.

By relationships between roots and co-efficients, (Viete’s relations), we get

$\alpha+\beta+\gamma=-\frac{3b}{a}$ and $\alpha\beta+\beta\gamma+\gamma\alpha=\frac{3c}{a}$ , and $\alpha\beta\gamma=-\frac{d}{a}$

Now, $\gamma=\frac{\beta\gamma-\alpha^{2}}{\beta+\gamma-2\alpha}=\frac{\frac{\alpha\beta\gamma}{\alpha}-\alpha^{2}}{(\alpha+\beta+\gamma)-3\alpha}=\frac{-\frac{d}{a\alpha}-\alpha^{2}}{-\frac{3b}{a}-3\alpha}=\frac{d+a\alpha^{3}}{3\alpha(b+a\alpha)}$ , that is,