The Quartic Formula

by Curtis Bright

It is a celebrated mathematical theorem that a formula exists which can solve general quartic equations. The formula consists of additions, subtractions, multiplications, divisions, and extraction of nth roots. Furthermore, no such formula exists for general quintic (or larger degree) equations.

Because of the complexity of the quartic formula it is almost never completely written out in full like the simpler quadratic formula is. I figured that attempting to print it using a normal font size and formatting it like a typical equation would require a poster-sized piece of paper. So I decided to put it on a poster!

I was able to find other webpages which wrote out the quadratic formula in full, but found them to suffer from some drawbacks. Most importantly, they don't always work; there are valid inputs which result in a division by 0. Also, they use four separate equations instead of one, and aren't formatted very aesthetically, at least to me.

My goal was to write it as a single formula that would work for all inputs. But I also wanted to make it as simple and self-contained as possible—I didn't want to add a clause like "choose values for the radicals which satisfy such-and-such equation". In short, I wanted to style the quartic formula like how the quadratic formula usually is. Ideally, the four solutions would be specified by two ± signs, similar to how the quadratic formula uses a single ± sign to specify two solutions.

There was a constant tension between accuracy and consiseness, but I'm satisfied with how it turned out. The result is a single formula which gives all roots of all quartic equations with a simple rule for selecting the radical values and ± signs. The ugliest part is a long expression (which makes up about one sixth of the formula) using the sgn function just to get the sign of the last radical correct.

If you just want to see the formula, I've posted it online along with the formulas for solving polynomials of smaller degree.

I'm also making it available as a PDF poster (with source available), a basic webpage, a MathJax webpage (which can take a few moments to load), a MathML webpage (which isn't supported in all browsers), and ASCII text. I've also included a mathematical derivation of the formula, in case you're interested in the inner details. And since the quartic formula relies on the cubic and quadratic formulas, I'm also making the above available for those formulas as well. It's interesting to see how the same general methodology which solves the quartic can also be used to solve the cubic and quadratic.

For typesetting the poster I used TeX (of course) and a half-A0 paper size. Although the Computer Modern fonts can be generated at any necessary size, Type 1 versions are only avaiable at a fixed number of sizes, so I had to make some substitutions and scale the fonts as necessary. The Type 3 PDFs use fonts of exactly the right size, but contain no hinting and will look poor on-screen—but they might be useful if you decide you want to print the poster.

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Poster Source