The Quadratic Equation
I was in middle school taking Algebra I when someone, probably another student, said I should open the textbook to a page about 3/4 of the way into the book. They pointed to a terribly complicated equation and said "you'll have to learn THAT". Here is the equation:
I was 13 at the time, and wise to the ways of the world and of math, so I knew immediately that this person was trying to put something over on me. Oh sure, this messy-looking equation might have been the solution to some particular (complicated) homework problem, or it may even have been an intermediate step on the way to a more elegant solution, but there was NO WAY that something this "messy-looking" was actually a meaningful and important piece of mathematics. I'd been going to school for over 8 years now, and every piece of mathematical insight I had seen had been strikingly simple and elegant, never a complicated mess like this.
Well, live and learn I suppose.
The quadratic equation is, of course, the solution to a fundamental question, and it has no other form I know of which is much simpler. The world is just a bit messier than I had believed in those sheltered days.
Since then I have come to believe the following:
(1) The "messy" is far more common than the "simple" or "elegant". Just as nearly-all functions are not expressible in closed form and nearly-all real numbers are irrational, nearly all problems have very difficult or messy solutions. I'd love to be able to state this precisely, perhaps even prove it, but I'm not quite sure how to go about it.
(2) An astonishingly large portion of the really important and interesting problems have simple and elegant answers. The Navier-Stokes equation is the exception - most things we care about CAN be solved simply. Why this should be is a truly deep question.