Solving a Cubic via the Trisection of an Angle

coefficient of x

-6

constant term

3

step

1

2

3

4

5

6

7

step 1 in solving

3

x

-6x+30 for x

The cubic is

3

x

+ px+q

3

x

-6x+3, so p-6 and q3.

The discriminant is -4

3

p

-27

2

q

= 621 > 0, so there are three real roots and we can proceed.

We wish to find the three real solutions to

3

x

-6x+30.

First, compute the number ϕ

3

q

2

-p

p

-

3

4

2

.

As the discriminant is positive, -1≤ϕ≤1, and therefore ϕ is the cosine of some angle θ.

This Demonstration shows a geometric solution to the equation

3

x

+px+q=0

, where

p

and

q

are real in the case where the equation has three distinct real roots (i.e. has a positive discriminant). It is based on Viète's trigonometric solution of the cubic that constructs and trisects a particular angle associated with the given cubic equation.