Express roots of equation $acx^2-b(c+a)x+(c+a)^2=0$ in terms of $\alpha, \beta, $

As given by me in comment:

let $g(x)=acx^2-b(c+a)x+{(c+a)}^2=0$ and $f(x)=ax^2+bx+c$

we see that $$g(x)={(c+a)}^2\cdot \frac{1}{c}\cdot f(\frac{-cx}{c+a})$$ thus $g(x)=0$ implies $$\frac{-cx}{c+a}=\alpha ,\beta$$ now use vieta and rearrange .....

Rewrite the equation $acx^2-b(c+a)x+(c+a)^2=0$ in the form

$$c\left(1+\frac ac\right)^2\frac1{x^2} - b\left(1+\frac ac\right)\frac1{x}+a=0 $$ and compare with the given equation written in the form $$\frac c{x^2} -\frac bx +a=0$$

to establish the relationship between their roots $$\left(1+\frac ac\right)\frac1{x_1}=\frac1{\alpha},\>\>\>\>\>\left(1+\frac ac\right)\frac1{x_{2}}=\frac1{\beta} $$ which, with $\frac ca = \alpha \beta$, leads to the roots $$x_{1}= \alpha +\frac1{\beta} ,\>\>\>\>\> x_{2}= \beta +\frac1{\alpha} $$