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!

Solve the two equations to find
$$ay^2+by+c=0\implies y=\dfrac{-b\pm\sqrt{b^2-4ca}}{2a}\ \ \ \ (1)$$
$$cax^2-b(c+a)x+(c+a)^2=0\implies x=(c+a)\cdot\left(\dfrac{b\pm\sqrt{b^2-4ca}}{2ca}\right)\ 
 \ \ \  (2)$$
Considering the opposite signs of the roots,
$$\dfrac yx=-\dfrac c{c+a}$$
Considering the same signs of the roots,
$$\dfrac yx=-\dfrac c{c+a}\cdot\dfrac{b+m\sqrt{b^2-4ca}}{b-m\sqrt{b^2-4ca}}\text{ where }m =\pm1$$
So, the ratios are not the same!
