Built on Facts

The quadratic formula. With the exception of the Pythagorean Theorem, it's probably the single most common mathematical formula people carry from high school. It's not a function as such, it's something that solves a function. Let me give an example:

Pick a number x, square it, add twice x to that, subtract 3 from that. You might want to figure out what x makes the answer equal to zero. It's the kind of thing we have to do in physics all the time. In this case, the answers (and you can find them by several methods) happen to be -3 and 1. Plug them in and you'll see that you end up with 0 as the answer.

In general if you have a quadratic equation like this:

The particular x that solves this equation (and there will be two of them - though they might be complex numbers, and they might be the same number, long story...) will be given by the quadratic formula:

Not so bad. But what if we're trying to solve a cubic equation, like this?

Well, just like the quadratic equation had two solutions, the cubic equation will have three. Here is the first:

Can you even read that? It's pretty tough for me. There's two more of course, for the other two solutions.

What about the quartic equation? It looks like this:

We can do it too. I'll spare you the horror, but it's about as much uglier than the cubic equation than the cubic equation is uglier than the quadratic equation. And there's four solutions, not just three.

What about the quintic equation? The one with an x^5 term?

Surprisingly - or perhaps mercifully - there isn't a solution. Some mathematicians much smarter than me proved it in 1824. At least there isn't a solution in anything with just fractions, powers, and roots like the ones we've seen here. But sometimes you really do need the actual numbers that solve quintic or even higher equations. Fortunately there's tricks we can do. These days computers automate most of them, but even before computers it was possible to get a decimal approximation to the solutions of these equations without needing an exact algebraic solution. And to be totally honest, that's pretty much all we ever do for any polynomial worse than the quadratic.

It's important to remember that there really are real live numbers that solve these polynomials of 5th and higher order, just not numbers we can express in terms of the standard arithmetic operations. But we're physicists. Experiments pretty much always give us decimal approximations of reality anyway, so we're used to it.