This is a really basic question and to be honest I feel ashamed to be asking this when I'm in precalculus and trigonometry right now.
When I was younger, I was taught that $\sqrt{x^2}$ was equal to $\pm x$. However, during this course I've inputted the negative before but was told that it was wrong, and that instead it was only the positive solution, not the negative as well. I tried contacting the professor numerous times but they haven't responded in weeks so I figured I'd come to here. Any help would be greatly appreciated.
You shouldn't be taught that $\sqrt{x^2}=\pm x$. The definition of $\sqrt{x^2}$ is $|x|$. $\sqrt{}$ is a function and can have only one value.
What you learned before, is that there are two solutions to $x^2-a^2$ - namely $x=\pm a$. Those can be written as $x^2-a^2=0 \implies x=\pm\sqrt{a^2}=\pm a$. But $\sqrt{x}$ is a function and only has one value (the positive one if $x\geq 0$, the primitive root otherwise).