Cardano's Method

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The Microworld…


The Story…


We come now to the center of this book. We will present a simple and transparent heuristic for solving cubic equations that will make the strategy of Cardano’s Method easy to remember.  Suppose we have a cubic equation to solve which has the form:

 

The coefficients A,B, and C here may be complex numbers.

We first make the change of variable that translates the curve so that the abscissa of the inflection point is 0. This is called completing the cube. Thus, letting  we now attempt to solve the equation

 

but write this in the simpler form:  where

 

We assume in the sequel that  since that reduces easily to the quadratic case. Now, we know that the sum of roots of this cubic is 0. And we know that there is a cubic number  with the property that  

Once we find this number, we will have solved the cubic equation: . The solutions will be: , or

 

            where  and of course,

      The solutions then to the original cubic equation:   are then of the form:   for these u.

Before we do that, however, let us explain the short-cut mentioned at the beginning of the book. Instead of attempting to solve the traceless cubic  directly for a complex solution, we look instead for a a cubic number w in the form  which is solution for . Then, given b and c, the complex solutions to  will be

 

      That’s all there is to Cardano’s method!  We will see how this heuristic will produce solutions to the cubic once we do the calculation outlined above. That calculation will justify this shortcut.

Now for cubic number ,  means that

 

And expanding this, using the identity: , we obtain the triple of equations:

 

Since we are assuming that  it is clear that a pair  satisfies this triple of equations if and only if it satisfies the reduced pair of equations:

 

But we can solve this pair, because the equation  simply means that the following complex polynomials must be equal:

 

We know that we can make them equal, because every monic cubic polynomial is (as we saw) the characteristic polynomial of some cubic number. And so this means that we must solve the pair of equations for b and c:

 

to obtain the desired cubic number  associated to .  Now we see that, because we assumed  that  and therefore  any cubic number w in the form  which is solution for  is associated to  in the above sense.

Thus, Cardano's method reduces to the steps:

1.      Transform   by completing the cube

2.      Writing  , if , find traceless cubic number  with

3.      The solutions are:

4.      The solutions then to the original cubic equation:   are then of the form:   for these u.

 

For step 2, we solve the pair of equations,

 

            by first solving the pair

 

            We get a quadratic equation to solve, and we find

 

Since  remains unchanged if we multiply b or c by a cube root of unity, and since  may differ from  only by such a factor, we are free to multiply b or c by that factor to satisfy . Thus the roots are

as we have said. And we obtain the original roots by adding  (the average of the roots) to u. Notice that b and c are in general complex numbers. It is obvious that if c is the conjugate of b, then these roots are all real, and in fact, they have the simple representation:

 

This is the case of an Hermitian cubic number, and we will study this situation more closely in the final chapter of the book. On this exploration page, we put the Expert System to work solving equations.

In the field labeled: Equation, type an equation. It may have more than one variable. In the variable field, type the variable to solve for. Then press the Solve button. With the Explain check set, the Expert System will explain its steps as it attempts to solve the equation.

Each time it pauses to explain its step, press Yes to continue. The Expert System uses Cardano's method to solve the equation, explaining its steps! You may find it easier, however, to use the short-cut that we have explained above, and to compare the Expert System’s answer with yours.