Is $\sqrt{x^2} = x$?

Question

Does the $\sqrt{x^2}$ always equal $x$? I am trying to prove that $i^2 = -1$, but to do that I need to know that $\sqrt{(-1)^2} = -1$. If that is true then all real numbers are imaginary, because an imaginary number is any number that can be written in terms of $i$. For example, 2 can be written as $i^2 + 3$. Does this work or did I make an error?

Answer 1

Not always. $\sqrt{(-1)^2}=\sqrt{1}=1\neq -1$. In general $\sqrt{x^2}=|x|$