One of the most mind-boggling acheivements in the history of human thought is the proof that the fifth degree equation is unsolvable. It's one thing to solve a hard equation, and another thing to prove that you can't solve it. It's a fascinating story that isn't well told.
We torture high school students with the solution of the quadratic equation. Most people will remember it as the hardest formula they ever had to memorize. This completely misses the essential point about the solving the quadratic: the equation is the simplest possible equation that makes sense. Reduced to its essence, the quadratic formula basically says: the solution of the quadratic equations is a number plus the square root of a number:
A fascinating thing happens when we move on to the cubic equation. Here again, the standard treatments are horribly misguided, by my way of thinking. Normally they take you through some very difficult algebra whose end result is a horribly intimidating mass of symbols. "This", they tell you, "is the solution of the cubic equation. The fourth degree is even more complicated."
Instead of emphasizing the complexity of the solutions, the correct approach is to recognize their simplicity. We look for the simplest possible solution of the cubic equation: and by analogy with our simplified solution to the quadratic equation, we look first to the obvious: a number plus the cube root of a number:
In particular, since every quadratic equation is also a cubic equation (albeit with a leading coefficient of zero), we need a formula that allows for taking square roots as well as cube roots. The least complicated formulas that fill these minimal requirements would be something like this:
In fact, it's not too hard to verify that formula we've written is the solution of a fairly simple sixth-degree equation in alpha. You can generate the exact equation by cubing both sides, collecting terms, and then squaring both sides again. But that's not where we want to go with this.
The real question is: how do we write a solution for the cubic equation that is powerful enough to do the job, and at the same time doesn't generate too many solutions?
Let's take that up when we return.