 So you know that there are general formulas for solving linear equations and for quadratic equations. Cardano's formula is a general formula that will allow us to solve a cubic equation. So the simplest version of it begins as follows. If I have a cubic equation of the form x cubed plus px equals q, this is a depressed cubic, there's no quadratic term, then my solution is going to be x equal to u minus v, where u and v are two new variables that satisfy the following relationship. u cubed minus v cubed is q, that's our constant term, and 3u v is p, that's our coefficient of x. This one's actually fairly straightforward to prove, we can do this by a direct substitution. Here's my original cubic equation. If I let x equal u minus v, then that expands out as follows, and then I can take advantage of the relationship u cubed minus v cubed equals q, 3u v equals p, and expand and simplify and end up with a true statement, which says that x equals u minus v, u cubed minus v cubed equals q, and 3u v equals p are going to in fact give us a solution to the original equation. So here's an example of how that might work. So let's say the equation x cubed plus 6x equals 20, this is actually Cardano's original problem, and so I can set up a system of equations, and that system again, the difference of the two cubes is equal to the constant term, and 3 times the product of u and v is equal to the coefficient of x. Now this is a fairly standard system of nonlinear equations in two variables, and we can actually solve this using any number of methods. But let's take a look at how Cardano solved this method, because there's some particularly interesting insights that result from it. So the first thing I'll do is I'll solve for u v by dividing both sides by 3, and then I'll cube both sides there. So I get u cubed v cubed equals 8, and the reason this form is interesting is if I think about u cubed and v cubed as being variables m and n, then what I have here is I have the difference of two things, and the product of two things, and this is a very classic form of an algebra problem. If you have the difference of two things and the product of two things, can you find what the things are? And so we can do that as follows. The reason that we found this is that we can solve for the sum of the two things, and so we'll do that by taking our u cubed minus v cubed, our difference, and squaring it, that'll give us u to the 6 minus 2 u cubed v cubed plus v to the 6 equals 400, and if I add 4 u cubed v cubed, so let's find out what that is. That's 32. If I add 4 u cubed v cubed, then the left-hand side simplifies as a new expression, which you should recognize as the square of the sum of u cubed and v cubed, and that tells me that u cubed plus v cubed itself is going to be the square root of 432, or simplifying it to root 108. And so what I have now, I have the difference of two things, I have the sum of two things, and that means I can find the two things individually very easily. So I'll go ahead and write down that second equation there. Now if I add these two equations, the v cubed terms drop out, and I'm just left with the u cubed. So that gives me 2 u cubed 20 plus 2 root 108, and solving for u cubed, and then solving for u is just going to be the cube root 10 plus square root 108. So I have that first term, and then similarly, if I subtract the two equations, if I take this equation here and subtract u cubed minus v cubed, then the u's drop out, my v cubed minus a negative v cubed becomes 2 v cubed, and the constant terms become what they are, and I can solve for v, and so I get v equals cube root of square root 108 minus 10, and again Cardano's rule says that x is the difference between these two. So my solution x equals cube root 10 plus root 108 minus cube root 108 minus 10.