<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] We look at the equation 
$$ax^2-bx(x-1)+c(x-1)^2=0.\tag{1}$$
If $a\ne 0$, then $1$ is not a solution, so Equation (1) is equivalent to 
$$a\left(\frac{x}{x-1}\right)^2-b\frac{x}{x-1}+c=0.\tag{2}$$
If we put $y=-\frac{x}{x-1}$ then the solutions of (2) are the solutions of $ay^2+by+c=0$. It follows that 
$$\frac{-x}{x-1}=\alpha\quad\text{or}\quad \frac{-x}{x-1}=\beta.$$
Now we can easily solve for $x$. 
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