 We can also solve cubic equations using a somewhat modernized form of Cardano's rule, although it turns out the modernized form is actually slightly more complicated. So again, we have our cubic equation, x cubed plus 6x equals 20, and again, Cardano's rule will give us a system of equations in U and V. This time, we'll use substitution. So let's solve that second equation for V, and now I have V equals 2 over U. I'll substitute that into my first equation, and I'll end up with a 6th-degree equation, which is actually reducible to quadratic if I use the substitution y equal to U cubed. So rewriting that little bit, that's U cubed squared minus 8 equals 20 U cubed. y equals U cubed allows me to rewrite this as a quadratic equation, and I can apply the quadratic formula to get my solutions y equals 10 plus or minus root 108. And one important thing to note here is we actually get two solutions here, and so you might wonder if the fact that we have two values of y is going to give us two values of U, and then that's going to give us two different solutions, x equals U minus V. Now let's take a look at it. So let's take the first solution, the positive solution y equals 10 plus square root 108 and solve for U and V. So remember, y is U cubed. That means U is the cube root of y. So if I'm taking that first positive solution, 10 plus root 108, there's my value for U. I can find the value for V a lot more easily by using the first equation. The second equation is going to be very messy. The first equation actually isn't too bad. Remember, y is U cubed. So this expression here is going to be the value of U cubed. So that tells me V is U cubed minus 20, and again I know that U cubed y is 10 plus root 108, and that simplifies, and V is the cube root, and again x is the difference U minus V. How about that second solution, that 10 minus root 108? So if y equals 10 minus root 108, I can go through the same process. U is still the cube root of y, it is going to be this. V cubed still U cubed minus 20, 10 minus root 108 minus 20, and that gives me what V is equal to, and again x is the difference of U minus V, and it seems that this is actually a different solution. However, remember that the cube root of a negative quantity is the same as negative the cube root of the quantity. So it turns out these two solutions are the same. This second term here is really plus cube root 10 plus 108 minus root 108 minus 10 cube root. So this solution here works out to be exactly the same as this solution here. So there's actually not two distinct solutions here, there's just the one solution.