Does $a^2 = b^2$ imply $|a| = |b|$?

Question

This seems like a rather obvious fact, but I can't figure out how to prove (or disprove) it.

Suppose $a, b, \in \mathbb{R}$, and $a^2 = b^2$. If I take square roots, I get $a = |b|$ and $b = |a|$. I want to conclude that $|a| = |b|$, and this seems to be rather obviously true, but I can't seem to get it via substitution. Perhaps the solution is to consider cases and prove that $|a| - |b| \geq 0$ and $|b| - |a| \geq 0$.

Answer 1

$$a^2=b^2$$

$$a^2-b^2=0$$

$$(a-b)(a+b)=0$$

$$a=b \textrm{ or } a=-b$$

In either case: $$|a|=|b|$$

Answer 2

$a^2=b^2\implies\sqrt{a^2}=\sqrt{b^2}\implies|a|=|b|$