I've been helping a high school student with his complex number homework (algebra, de Moivre's formula, etc.), and we came across the question of the "usefulness" of "imaginary" numbers - If there not real, what are they good for?
Now, the answer is quite obvious to any math/physics/engineering major, but I'm looking for a simple application that doesn't involve to much. The only example I've found so far is the formula for cubic roots applied to $x^3-x=0$, which leads to the real solutions by using $i$.
Ideally I'd like an even simpler example I can use as motivation.