Ratio null equations

Let the roots of the equation $ax^2+bx+c=0$ be $\alpha$ and $\beta$ . Then $r = \dfrac \alpha \beta$ . By Vieta's formula , we have $\alpha + \beta = - \dfrac ba$ and $\alpha \beta = \dfrac ca$ . Then,

$\begin{aligned} (\alpha + \beta)^2 & = \frac {b^2}{a^2} \\ \frac {\alpha^2 +2 \alpha \beta + \beta^2}{\alpha \beta} & = \frac {b^2}{a^2} \times \frac ac \\ \frac \alpha \beta + 2 + \frac \beta \alpha & = \frac {b^2}{ca} \\ r + 2 + \frac 1r & = \frac {b^2}{ca} \\ \frac {r^2+2r+1}r & = \frac {b^2}{ca} \\ \frac {(r+1)^2}r & = \boxed{\dfrac {b^2}{ca}} \end{aligned}$