Find the roots of $acx^2-b(c+a)x+(c+a)^2=0$

Question

If $ax^2-bx+c=0$ roots are $\alpha, \beta $ then find the roots of $$acx^2-b(c+a)x+(c+a)^2=0$$ from $\alpha, \beta$.

I need to find the best way to anwser this question

my answer

Say $\gamma, \delta $ are the roots of given equation then $$\gamma\cdot\delta=\frac{(c+a)^2}{ac}=(\alpha+1/\beta)(\beta+1/\alpha)$$ and $$\gamma+\delta=\frac{b(c+a)}{ac}=(\alpha+1/\beta)+(\beta+1/\alpha).$$ So roots are $$(\alpha+1/\beta),(\beta+1/\alpha)$$ this method is less powerful (it is hard to find the relation of $\alpha,\beta$ and $\gamma, \delta$)

Can anyone give me a different answer to this question? Thanks!!

Answer 1

Your approach is just fine and also very general. You can also try this method:

Try to form given equation into the old form $ay^2-by+c =0$ and deduce the relation of $x$ and $y$. First divide by $(c+a)^2$:

$$ac(\tfrac{y}{c+a})^2-b (\tfrac{y}{c+a})+1 = 0$$

Now we need a $c$ in place of $1$. So multiply throughout by $c$:

$$a(\tfrac{yc}{c+a})^2-b (\tfrac{yc}{c+a})+c = 0$$

Thus we have found out that $x = \tfrac{yc}{c+a}$ or that $y = x(1+\tfrac{1}{P})$ where $P$ is the product of roots of original quadratic.