<s>[INST] Finding the roots of a different Quadratic equation from the roots of a Given Quadratic equation The Question:

If $\alpha$ and $\beta$ are the roots of the equation $ax^2+bx+c=0$...
Then find the roots of the equation $ax^2-bx(x-1)+c(x-1)^2=0$


My Attempt:
The new equation can be made into a quadratic as:
$$(a-b+c)x^2+(b-2c)x+c=0$$
Now $$\text{Sum of roots}=\dfrac{-b}{a} = \dfrac{2c-b}{a-b+c}$$
And $$\text{Product of roots}=\dfrac{c}{a}=\dfrac{c}{a-b+c}$$

But I don't seem to be going anywhere with the way I'm proceeding
Please Help! Thanks!
 [/INST] hint: $\dfrac{2c-b}{a-b+c} = \dfrac{2\dfrac{c}{a}-\dfrac{b}{a}}{1-\dfrac{b}{a}+\dfrac{c}{a}}=\dfrac{2\alpha\cdot \beta+(\alpha+\beta)}{1+\alpha+\beta+\alpha\cdot \beta}$, can you continue?
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