Can anyone help me simplify this statement:
$$x = \frac{(1-a)^2}{(1-xa)^2}$$
I multiplied the denominator by $x$ to get rid of the fraction, but I cannot figure out how to manipulate it to reach the common $(x-z)(x-y) = 0$. Any help or suggestions would be greatly appreciated.
Note that $x=1$ is a solution.
Taking $(1-ax)^2$ to the L.H.S and converting this into a cubic gives: $$a^2x^3 - 2ax^2 + x - (1-a)^2 = 0$$ We can factor out an $(x-1)$ to give:
$$a^2x^3 - 2ax^2 + x - (1-a)^2 = (x-1)(a^2x^2 + (a^2-2a)x + (1-a)^2)$$
You can now look for other solutions by solving the quadratic.