The Renaissance Solution of the Cubic Equation

Let us begin with the equation of Del Ferro

x3+ mx = n

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

3ab=m and a3 - b3 = n

then

x3+ 3ab x = a3 - b3.

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

(a - b)3+ 3ab(a - b)= a3 -3 a2b + 3 a b2 - b3 + 3a2b - 3ab2

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

(a - b)3 + 3ab(a - b)= a3 - b3.

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.

b = m/(3a)

Plug this into the other equation to get

n = a3 -( m3/(27a3))

Then

n a3= a6 -( m/3)3

and

a6 - n a3 -( m/3)3 = 0.

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

z2 - n z -( m/3)3 = 0.

with solution

z= (n/2) + (1/2)(n2 + 4(m/3)3)1/2

or equivalently

z= (n/2) + ((n/2)2 + (m/3)3)1/2

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

a=( (n/2) + ((n/2)2 + (m/3)3)1/2)1/3

If we take

b=( - (n/2) + ((n/2)2 + (m/3)3)1/2)1/3

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

x = ( (n/2) + ((n/2)2 + (m/3)3)1/2)1/3 - ( - (n/2) + ((n/2)2 + (m/3)3)1/2)1/3

Let's use this formula to try to solve

x3+ 24x = 56

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

a3 = 28 + (784 + 512)1/2= 28 + 36 = 64

On ther other hand,

b3 =- 28 + (784 + 512)1/2 = -28 + 36 = 8

and a - b = 641/3 - 81/3 = 2, which is a solution:

23 + 24(2) = 56.

Observe that

x3+ 24x - 56 = (x-2)(x + 2x +28)

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)

x3 -18 x = 35



To continue see The Renaissance Solution II