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.

Your methodology isn't wrong. You just need to modify it a bit. Before you divide both sides by $x$, you must check what  happens when $x = 0$; that is, you must check whether or not $x=0$ is a solution. Once you have taken care of that, you are free to consider what happens when $x \ne 0$. In particular, now you can divide both sides by $x$.
(1)    $x^2 = x$
(2)    Clearly $x = 0$ is a solution. 
(3)    Now supose that $x \ne 0$. Then we can multiply both sides by $\dfrac 1x$.
(4)    We get $x = 1$.
(5)    The solution set is $x \in \{0, 1\}$.
