..I've done math problems on some paper, and there are no solutions given.

1. Simplify

(My) solution:

Two solutions?

Now, what if you've "gained" some (and which?) solution by squaring the original?

Mr F says: You have to go back to the original expression and test each potential solution. Clearly therefore discard the positive solution.

[snip]

3. Solve

Again, what if you've "gained" some solution by squaring the original? What are considered best practises in solving such?

Mr F says: You have to test each potential solution in the original equation. Both solutions satisfy the original equation so you keep both.