I would like to convince a high school student that $i$ is a number, broadly put. I'm not going to define what I mean by "number" unless he asks, but I just want to convince him that it's somehow meaningful and useful. Part of this I've already done by showing the geometric interpretations of complex numbers as points on a complex plane with geometrically sensible and interesting interpretations of addition and multiplication. That's probably the most important part.
But I'd like to take it a step further if that's not asking too much, and I know that complex numbers started getting studied by studying cubic equations, particularly because we know that the curve $y=x^3$ must intersect any line $y=mx+b$ somewhere. Simple suggestive graphs convince us of this. So can someone give me an example of such a cubic equation that is most easily solved using complex numbers?
I'm not exactly sure what I have in mind for that (Will I ask him to use the complex root with polynomial division to find the other roots? Or is there some other assignment that is perhaps easier for a high schooler to grasp?). Any recommendations would be appreciated.
Icing on the cake is if the equation had some kind of meaning in Physics or Geometry, like modeling the volume of a certain box.
[Edit: Perhaps I should elaborate with how I came to be focused on getting some kind of an example with a cubic. I've read in some history of Math sources that the study of complex numbers arose from the study of certain cubics and the search for a general solution. I figure, if the study of cubics is what compelled people in history to take complex numbers at least a little seriously (and then later on, take them even more seriously) then it could serve as a helpful example. However, when I looked at the cubic and the method of solution Bombelli used to find the solutions of $x^3=15x+4$, it was clearly too complicated to give most of the high school students I work with. So I was wondering if there was some similar type of magic that was simple enough for an introductory lesson in complex numbers. But I'm starting to guess than the answer is no.]