The Cardano's formula (named after Girolamo Cardano 1501-1576), which is similar to the perfect-square method to quadratic equations, is a standard way to find a real root of a cubic equation like
We can then find the other two roots (real or complex) by polynomial division and the quadratic formula. The solution has two steps. We first "depress" the cubic equation and then solve the depressed equation.
To depress the cubic equation, we substitute
Then we get
Note that the above equation has been depressed in such a way that it is without the term. Let's try this on an example.
Depress the following equation:
Substituting the equation is depressed to
The next step is to solve the depressed equation of the form .
We first substitute
Then we solve for
Find the real root of the cubic equation
From the above example, we know the given equation is depressed into Then we substitute
From we have substituting which into gives
Multiplying throughout by , we obtain the quadratic equation for as follows:
Solving for and then , we have
It can be shown that, whether we take the positive or negative root of we will get the same value for . We shall only consider the positive here.
From we have
Therefore, the real root of the original cubic equation is
Note: We check that is the only real root of the cubic equation because can be factorized as
The cubic polynomial has solutions
where and in turn .
After depressing the equation and making it monic by dividing by we get
Now consider the identity . If we make it match with the equation, we get
Cube both sides of the second equation to get . Now, by Vieta's formula, the polynomial will have roots and , so let's solve it using the quadratic formula:
Notice that the system of equations is symmetric in and , so the order we choose doesn't matter, and the value of will be the same. So
where and is any primitive root of unity. We see that then we have 9 combinations for the value of , but only 3 of them work. By looking at the second equation, we see that must be a multiple of 3, so and our solutions are
We choose and , so
Undo the change and we get our desired solutions.
Find the roots of the following polynomial equation using Cardano's method:
It is given that
Compute the roots: