Why does basic algebra provide one value for $x$ when there should be two? I have the equation $x^2=x$.
If I divide $x$ from both sides I get $x=1$.
Yet clearly $x$ can also equal $0$.
What step in this process is wrong? It seems to me that there's only one step. And isn't dividing the same thing from both sides a valid step?
I hope this isn't a stupid question because I feel dumb asking about something so basic.
EDIT: To clarify, what I'm looking for is not only an explanation for why my methodology is wrong, but also a better methodology that will keep me from missing possible solutions in the future.

You cannot divide by $x$ if $x=0$ in the first place. That's why if you want to divide by $x$, you take cases. If $x=0$, we have a solution. If $x\neq0$, then we get $1$.
