Most of us know the solution formula of the quadratic equation

by heart. But do you know the formula for the cubic equation by heart? No? Nor do I but I know that it exists and where I can find it. Only recently, I read a little bit about the history of the cubic formula, which is quite intruiging. While the quadratic formula was already known before the Greeks, the solution to the cubic equation was found only after many attempts in the 16th century. At that time, people (we wouldn’t call them scientists today) challenged each other with mathematical problems. For instance, Gerolamo Cardano found a method to solve the socalled depressed cubic

which he kept secret for a long time, and he used his secret knowledge to challenge other people with that problem. Cardano was an Italian polymath, physician, mathematician, astronomer, and astrologer of the Renaissance. He was born in Pavia, Italy, in 1501 and died in Rome in 1576. He was a member of a distinguished family of lawyers and was himself trained in law, but his interests were primarily in mathematics, science, medicine, and … passionate gambling! Finally, he published his knowledge in his famous book “Ars magna” (Latin for great art), by which he meant algebra or solving equations. But the depressed cubic equation is not the general form

which also involves a quadratic term. So, another idea was needed to crack the general cubic. It’s actually quite surprising that the discoverer of the solution to the depressed cubic didn’t go all the way to the most general form because the final idea missing is simple compared to the ideas involved in solving the depressed form. But judge for yourself. Here, we will do the derivation with Python’s sympy. So, start a Jupyter notebook and define your symbols:

The strategy is to make substitutions to bring that into a form without a quadratic term (the depressed cubic). And then we will apply Cardano’s method.

The first step is to let the cubic term have a coefficient of 1. So we divide by 𝑎:

Now we try to get rid of the quadratic term by making a suitable substitution. That is, we substitute

and choose 𝑤 as we like.

We want the quadratic term to vanish, so we choose 𝑤 to satisfy

or

Now we define constants 𝑞 and 𝑝 as

With these abbreviations, we have arrived at the depressed cubic form

Note that all we have done to get to the depressed form is to make a dummy substitution and compare coefficients to get rid of the quadratic term. That doesn’t require genius. The next steps, however, need much more cleverness. This is Cardano’s method.

Starting with the depressed form, we make another substitution:

So we change one unknown 𝑦 for two unknowns 𝑢 and 𝑣. So we are free to choose one of them as we like.

We bring it into a more convenient form to isolate the cubic terms:

As I wrote above, we are free to choose a condition for u and v. Let’s choose it such that the term in brackets vanishes:

So we have

and also

Substituting for 𝑣³ gives

Now we are almost there. Making another substitution

turns our problem into a quadratic equation

And of course, we can apply the well-known quadratic formula here. So we get

and

Now we have our solution for the depressed cubic, called Cardano’s formula from y = u+v

It may look as if this yields 9 solutions instead of three, but the condition for 𝑢𝑣uv makes sure that there are really only three different solutions.

To wrap up, this is the recipe:

Starting with

Make the substitution

and calculate the constants

and

Then we can insert that into the expression for 𝑦 above and work back to get 𝑥.