Week 5 - Solving Cubic Equations (Doing)

Cubic Formula Derivation

The cubic equation is of the form:

x^3 + ax^2 + bx + c = 0

Cardano solved the cubic equation by doing these steps:

x = y - a/3

via the Tchirnhaus transformation, and this simplifies the equation to:

y^3 +py + q = 0

where 

p = b - (a^2)/3

and q = c - ab/3 + (2a^3)/27

then substitute y = u - v to get:

(u-v)^3 + p(u-v) + q = 0

and this can be written as:

(q - (u^3 - v^3)) + (u+v)(p - 3uv) = 0

With the two separate terms, we see that q = u^3 - v^3 and q = 3uv, and this solves the equation. 

As for the remaining terms in the equations for q, we can solve for u or v and use substitution to finish the equation.

These roots are solved using methods such as Newton’s Method. The numbers a, b, c, d, u, v, q, and p are real numbers.