How to justify $\pm$ is unnecessary on both sides with respect to $x_1^2 = x_2^2 \implies \pm x_1 = \pm x_2$

Question

So I taught one-to-one functions in my class today and totally embarrassed myself with my explanation of trying to explain why you don't need to put $\pm$ on both sides for $$x_1^2 = x^2_2 \implies \pm x_1 = \pm x_2.$$

It doesn't tell the whole picture exactly. Why though? I was thinking about the following counterexample: $$(-3)^2 = (3)^2,$$ but I just got stuck trying to explain further. Why is this bad practice exactly?

Answer 1

Your example says that squaring of the both sides gives an equivalent equality iff both sides have the same sign.

Actually, $$x_1^2=x_2^2\Leftrightarrow x_1=x_2\vee x_1=-x_2$$ because $$x_1^2-x_2^2=(x_1-x_2)(x_1+x_2).$$