When solving for the function $x^2=4$ we take the sqaure root of both sides $\sqrt{x^2}=\sqrt{4}$ then you get $x=\pm 2$, obviously this is because square both $2$ and $-2$ will get you $4$. My teacher said that when you simplify ($\sqrt{4}=$) alone the answer is simply $2$, He says that the this is also evident when you graph it and you get only one side of the graph (not the negative value -)
I asked my teacher today who wrote $\sqrt{x^2}=\pm\sqrt{4}$, my response to that was that the square root implies the $\pm$ and by adding it your essentially pulling that out of thin air, I understand that if your simply writing the answer as $\pm{2}$ the $\pm$ makes sense but should you have to include it when you leave it in radical form ($\sqrt{}$), as the square root should imply the ($\pm$)
Your teacher is right in the first paragraph and wrong in the second. The square root function always means the nonnegative root. $\sqrt{4} = 2$. The equation $x^2 = 4$ has two roots, $2$ and $-2$, which you can write as $\pm \sqrt{4}$.