Q.
If $\alpha, \beta, \gamma$ be the roots of the equation $ax ^3+ bx ^2+ cx + d =0$. To obtain the equation whose roots are $f (\alpha)$, $f(\beta), f(\gamma)$, where $f$ is a function, we put $y=f(\alpha)$ and simplify it to obtain $\alpha=g(y)$ (some function of $y)$. Now, $\alpha$ is a root of the equation $ax ^3+ bx ^2+ cx + d =0$, then we obtain the desired equation which is $a \{ g ( y )\}^3+ b \{ g ( y )\}^2+ c \{ g ( y )\}+ d =0$
For example, if $\alpha, \beta, \gamma$ are the roots of $ax ^3+ bx ^2+ cx + d =0$. To find equation whose roots are $\frac{1}{\alpha}, \frac{1}{\beta}, \frac{1}{\gamma}$ we put $y=\frac{1}{\alpha} \Rightarrow \alpha=\frac{1}{y}$
As $\alpha$ is a root of $a x^3+b x^2+c x+d=0$
we get $\frac{a}{y^3}+\frac{b}{y^2}+\frac{c}{y}+d=0 \Rightarrow d y^3+c y^2+b y+a=0$
This is desired equation.
If $\alpha, \beta$ are the roots of the equation $ax ^2+ bx + c =0$, then the roots of the equation $a(2 x+1)^2+b(2 x+1)(x-1)+c(x-1)^2=0$ are-
Complex Numbers and Quadratic Equations
Solution: