Question: Express the roots of the equation $q^2x^2-(p^2-2q)x+1=0$ in terms of those of $x^2+px+q=0$
My attempt:
Roots of the second equation are $\frac{-p±\sqrt{p^2-4q}}{2}$
Roots of the first equation are $\frac{(p^2-2q)±\sqrt{(p^2-2q)^2-4q^2}}{2q^2}$ which on simplification becomes $\frac{(p^2-2q)±p\sqrt{(p^2-4q)}}{2q^2}$
The discriminants of both the equations are the same.
My problem: I am unable to proceed further. I was trying to take the ratio of the roots and find a relation between them but this is not working out. I am also unsure whether the logic that i have used in my attempt is of any use.
Let the root of the second equation to $\alpha , \beta$
Then $$\alpha + \beta = -p$$
$$\alpha \beta = q$$
$$\frac{p^2-2q}{q^2}=\frac{(\alpha+\beta)^2-2\alpha\beta}{(\alpha\beta)^2}=\frac{\alpha^2+\beta^2}{(\alpha\beta)^2}=\frac{1}{\beta^2}+\frac{1}{\alpha^2}$$
$$\frac{1}{q^2}=\frac{1}{\alpha^2}\frac{1}{\beta^2}$$
Can you see the root for the first equation now?