How does this imply $x = z^2$?

Question

I read that given the equation:

$$ x + \frac{1}{x} = z^2 + \frac{1}{z^2} $$

, we can imply that $x=z^2$.

But that $x=z^2$ is not obvious to me... I know we can match the variables on the left hand side to the right hand side, but how do we show that this holds in general?

Answer 1

If you clear out the denominators: \begin{align*} x + \frac{1}{x} &= z^2 + \frac{1}{z^2} \\\implies \frac{x^2+1}{x} &= \frac{z^4+z^2}{z^2} \\\implies z^2x^2 + z^2 &= z^4 x + x \\\implies z^2x^2 + z^2 - z^4 x - x &= 0 \\\implies (z^2x-1)(x-z^2) &= 0 \end{align*} So $x=z^2$ or $x = \frac{1}{z^2}$.