The Renaissance Solution of the Cubic Equation

Let us begin with the equation of Del Ferro

Now suppose that we can find number a and b such that

then

The important fact to observe here is that x = (a - b) is a solution to this equation, since

and cancelling like terms on the right hand side of the equation we get

The only question we must answer to consider this a complete solution is whether the values a and b can be obtained. But the equations defining a and b are non-linear equations. This is a non-linear system of equations. This non-linear system of equations must be solved to obtain a solution. How do you solve a system of two equations and two unknowns? Substitution?

Let us solve for b in terms of a using the equation m= 3ab.

Plug this into the other equation to get

Then

and

resembles a quadratic equation. If we let z = a³, then we get a quadratic equation as follows

with solution

or equivalently

It is easy to see that a is the cube root of z, or

If we take

then we get that a³ - b³ = n. Therefore,

Let's use this formula to try to solve

Now n =56 and (n/2)= 28. On the other hand, m=24 and (m/3) = 8. Then (n/2)² is 784 and (m/3)³ is 512 and so we have

On ther other hand,

and a - b = 64^1/3 - 8^1/3 = 2, which is a solution:

Observe that

We can now solve for all the roots using the quadratic formula and we see that x = -1 + (1 - 28)^1/2, and x = -1 - (1 -28)^1/2 are not real roots.

Homework Assignment 3

Solve the equation using the formula to show that 5 is a root. (hint a³ is 27)