I am having trouble finding critical numbers, specifically finding the roots or zeroes of a function. Especially when it involves a fraction.
For example right now I have \begin{align*} f(x) &= x^2 - x - \ln x\\ f' (x) &= 2x - 1 - \frac {1}{x}\\ f' (x) &= 2x - 1 - \frac {1}{x} = 0\\ f' (x) &= 2x - \frac {1}{x} = 1\\ f' (x) &= 2x^2 - 1 = x. \end{align*} Here I multiplied by $x$ on both sides which seems to be okay to do but it does not give me the same answer as if I were to plug in a number, for example 11, into \begin{align*} f' (x) &= 2x - 1 - \frac {1}{x} = 0\\ f' (x) &= 2x^2 -x - 1 = 0\\ \end{align*}
I know these are basic math concepts I should have mastered by now but I can't figure this out. Shouldn't both those function be equal to eachother?