I am trying to solve the quadratic equation $x^2 + x + 1 = 0$. $x^2 = -1 - x $ $\iff x = -\frac{1}{x} - 1$, assuming $x\neq 0$.
Substituting that into the original equation gives $x^2 + (-\frac{1}{x} -1) + 1 = 0$
$x=1$ is a solution to this second equation, but it isn't a solution to the first. How did this extra solution arise? Which step wasn't reversible and why wasn't it?