Confused by this one cause when I do the discriminant trick, there are 4 solutions. Am I missing something? Maybe it’s the dollar sign throwing me off.

I think it’s the dollar sign.

No, that was a formatting bug which has been fixed.

To answer the OP’s question: can the LHS ever be negative?

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Ohhh, that’s right ok. So in order for this whole process to work, (where we re-write the equation twice, with one option as negative RHS), the equation has to be true in the first place before rewriting everything, right?

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Yes.

It is also good practice (and even necessary in some cases) to verify that the solutions you get when solving a problem are indeed valid. Spurious solutions are a thing.

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