I've got the equation $ x^2+y^2=2x $. By looking at the graph of that function, I know that it is equivalent to $ r=2\cos{\theta} $ (graph).
However, if I convert it by substituting in using the equations $ x = \cos{\theta} $ and $ r = \sqrt{x^2+y^2} $, I get $ r^2 = 2 \cos{\theta} $.
I'm sure it's something obvious, but what am I missing that gives me this incorrect answer?
You're missing the $r$ on the other side: $r^2 = 2r\cos \theta$.