I am working with a text book problem, trying to find the solutions to $x^3 = 2x^2 - x$, but I keep ending up with a different result than the book gives. I'm thinking the problem lie in getting the wrong square root towards the end, seeing as the text book gives different results for $x_1$ and $x_2$, but I can't find any mistakes.
$$x^3 = 2x^2 - x$$ $$\iff x^2 = 2x - 1$$ $$\iff x^2 - 2x + 1 = 0$$ $$\iff (x - \frac{2}{2})^2 = (\frac{2}{2})^2 - 1$$ $$\iff (x - 1)^2 = 0$$ $$\iff x - 1 = \pm \sqrt{0}$$ $$\iff x = 1 + \pm \sqrt{0}$$ $$\iff x = 1 + \pm 0$$ $$x_1 = 1 + 0 = 1$$ $$x_2 = 1 - 0 = 1$$