How to solve cubic and quartic equations I: quartics

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How to solve cubic and quartic equations I: quartics

July 17, 2026

We all know and love the quadratic formula. It says that the solutions to \(ax^2+bx+c=0\) are<br>\[<br>x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.<br>\]

However, high school algebra classes usually stop here. They do not, for instance, explain how one would solve an equation like<br>\[x^3 + x + 1 = 0\]<br>or<br>\[x^4 - 6x^3 + 7x^2 + 8x = 9.\]

The goal of this series of two articles is to explain how to solve cubics (equations of the form \(ax^3 + bx^2 + cx + d = 0\)) and quartics (equations of the form \(ax^4 + bx^3 + cx^2 + dx + e = 0\)), like the ones above.

Though, really, these two articles have a different goal: we want to introduce ideas of Galois theory to the reader. Galois theory, invented by Évariste Galois, is an incredible branch of mathematics, which systematically studies the symmetry of solutions to equations. For example, the fact that the quadratic formula involves the funny \(\pm\) (plus or minus) symbol is related to a certain symmetry of quadratic equations. By thinking about the symmetries that cubics and quartics have, one is led to the formula for solving them.

Historically, Galois was born long after the cubic and quartic had been solved. He was interested in using symmetry to prove that you can go no further: there is no formula you can write down which solves equations of the form<br>\[ax^5 + bx^4 + cx^3 + dx^2 + ex + f = 0\]<br>using only radicals (things like \(\sqrt{5}, \sqrt[3]{2}, \sqrt[4]{7},\) etc.), addition, multiplication, division, and subtraction. Galois proved this by thinking about the kinds of symmetries that the solutions to a quintic have, and then showing no formula could ever have that same type of symmetry.

But enough with the preamble; let's get into solving some equations!

Thinking about the quadratic formula

If we want to find a solution to the cubic or the quartic, it's useful to first ask how one might go about inventing the quadratic formula. And before we do that, it's worth asking: what does the quadratic formula even do, really?

The simplest sort of quadratic equation is something like \(x^2 = 17,\) where the solution is given by \(x = \pm\sqrt{17}\) (where, recall, the \(\pm\) means `plus or minus,' because both \(\sqrt{17} \approx 4.123\) and \(-\sqrt{17} \approx -4.123\) solve the equation \(x^2 = 17\)).

But something was hidden when we wrote \(\sqrt{17} \approx 4.123.\) The symbol \(\sqrt{17}\) just means "the (positive) number which squares to \(17.\)" So, when you ask someone for a solution to \(x^2 = 17,\) and they answer back "it's \(\pm\sqrt{17}\)", in some sense they've done nothing for you. The only reason we consider \(\sqrt{17}\) as being a valid answer to \(x^2 = 17\) is because the square root function comes up very often, and so humanity has developed many tools for quickly calculating it -- for example, your calculator probably has a button for it. People thus consider \(\sqrt{17}\) to be a valid expression to use in equations, and leave the task of numerically evaluating it to the reader of their solution.

Similarly, when one writes that<br>\[ax^2 + bx + c = 0\]<br>has solutions<br>\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},\]<br>one is essentially saying that if you know how to solve the equation<br>\[y^2 = b^2 - 4ac,\]<br>then you can solve \(ax^2 + bx + c = 0\) as well.

So, the quadratic formula is secretly a tool for turning one quadratic into a simpler quadratic. This reduction is more useful than it might seem, though: there are tons of algorithms for computing square roots (your calculator knows one, your computer knows one, and you could even learn one to execute with pen-and-paper for yourself!). So, the quadratic formula reduces an arbitrary quadratic equation to one for which we already have a solution.

People often joke that mathematicians are lazy, because this is incredibly common in mathematics: if you have some problem you cannot solve, like \(-x^2 - 2x + 1 = 0,\) then reduce it to some problem you can solve, like \(y^2 = 8.\) The quadratic formula is just a reduction of this type. Human science and engineering is built on the work of giants: if the people before you have already discovered something, it is a good idea to try and reframe your problem in a way that their past discoveries are applicable.

Vieta's formulas

We now have some understanding of what the quadratic formula is: it's a tool for turning quadratic equations into square root problems. But how does one actually invent the quadratic formula? For this, we turn to an idea of Vieta.

Suppose the quadratic equation<br>\[x^2 + bx + c = 0\]<br>has the two solutions \(x=\alpha\) and \(x=\beta.\) This means we can factor the quadratic as<br>\[x^2 + bx + c = (x-\alpha)(x-\beta).\]

The French mathematician Vieta noticed something amusing. If we expand out \((x-\alpha)(x-\beta),\) we get<br>\[(x-\alpha)(x-\beta) = x^2 - (\alpha + \beta)x + \alpha\beta.\]<br>Thus, if<br>\[x^2 + bx + c =...