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

Here's what I have tried so far,
I know that $\alpha+ \beta=\frac{-b}{a} $ and $\alpha \beta=\frac{c}{a} $
So I can express $b=-a(\alpha+\beta)$
$c=a.\alpha\beta$
Once I substitute for b and c in the equation I can get, $$\alpha\beta x^2+(\alpha+\beta)(\alpha\beta+1)x+(\alpha\beta+1)^2=0$$
I want to know whether there is any different approach other than this method?
Any hint is higly valued. thank you!

Let $r_1$ and $r_2$ be the roots of the equation
$$
a c x^{2}-b(c+a) x+(c+a)^{2}=0
$$
Then $$
\begin{aligned}
r_{1}+r_{2} &=\frac{b(c+a)}{a c}=\frac{b}{a}+\frac{b}{c}\\&=-(\alpha+\beta)-\frac{\alpha+\beta}{\alpha \beta} \\
&=-(\alpha+\beta)-\left(\frac{1}{\beta}+\frac{1}{\alpha}\right)\\&=-\left(\alpha+\frac{1}{\beta}\right)-\left(\beta+\frac{1}{\alpha}\right)
\end{aligned}
$$
and
$$
\begin{aligned}
r_{1} r_{2} &=\frac{(c+a)^{2}}{a c}=\frac{c^{2}+2 a c+a^{2}}{a c} \\
&=\frac{c}{a}+2+\frac{a}{c} \\
&=\alpha \beta+2+\frac{1}{\alpha \beta} \\
&=\left[-\left(\alpha+\frac{1}{\beta}\right)\right]\left[-\left(\beta+\frac{1}{\alpha}\right)\right]
\end{aligned}
$$
Hence the required roots are $\displaystyle -\left(\alpha+\frac{1}{\beta}\right) $ and $ \displaystyle -\left(\beta+\frac{1}{\alpha}\right) .$
