transformation z to w plane

Question

If im given $z^2-2z+4=0$ and I want to find that a transformation z to w such that $$\frac{4w^2}{(w-1)^2}-\frac{4w}{w-1}+4=0$$

I tried writing it as

$$w=f(z)=\frac{z-4}{z-1}\Rightarrow w^2 = \frac{(z-4)^2}{(z-1)^2}$$

Im not sure what to do from here on..

Answer 1

Try $$z=-\frac{2w}{w-1}+2.$$ It should help.

Let $\frac{w}{w-1}=x$.

Thus, $$z^2-2z+4=(-2x+2)^2-2(-2x+2)+4=4(x^2-x+1).$$

Answer 2

Edit
Michael Rozenberg's answer is better than mine, because I supplied a transformation from $w$ to $z$, when the problem asked for a transformation from $z$ to $w$.

$z = (2w)/(w-1)$ : this simultaneously satisfies the constraints on both terms that have a $w$ in them.

under this transformation, $2z = 4w/(w-1)$

and $z^2 = 4w^2/[(w-1)^2].$

Therefore, the constraint that $z^2 - 2z + 4 = 0$ continues to be satisfied in the $w$ plane.