$\text{ If }\alpha ,\beta \text{ are the roots of }a{x}^2+c=bx,\text{ then the equation }(a+cy{)}^2=b^{2}y\text{ in }y\text{ has the roots }$

$\begin{array}{l}a{x}^2-bx+c=0\\ \alpha +\beta =\frac{b}{a},\alpha \beta =\frac{c}{a}\\ \text{ Also, }(a+cy{)}^2=b^{2}y\\ \Rightarrow c^2{y}^2-\left(b^2-2ac\right)y+a^2=0\\ Divide by a^2\\ \Rightarrow {\left(\frac{c}{a}\right)}^2{y}^2-\left({\left(\frac{b}{a}\right)}^2-2\left(\frac{c}{a}\right)\right)y+1=0 \left[ {\left(\alpha +\beta \right)}^2-2\alpha \beta ={\alpha }^2+{\beta }^2\right]\\ \Rightarrow (\alpha \beta {)}^2{y}^2-\left({\alpha }^2+{\beta }^2\right)y+1=0\\ Divide by {\alpha }^2{\beta }^2\\ \Rightarrow y^2-\left({\alpha }^{-2}+{\beta }^{-2}\right)y+{\alpha }^{-2}{\beta }^{-2}=0\\ \Rightarrow \left(y-{\alpha }^{-2}\right)\left(y-{\beta }^{-2}\right)=0\\ \text{ Hence roots are }{\alpha }^{-2},{\beta }^{-2}.\end{array}$