Cubic Roots- Cardano and the Extended Quadratic Equation

Finding Cubic Polynomial roots using an abridged application of Cardano’s formula with the ‘Extended Quadratic Equation’

This post offers a 2 formula tool-kit to finding exact Cubic Polynomial roots using Cardano’s formula for 1 root and the Extended Quadratic Equation for the other 2. It’s Find 1 get 2 free!

While 3 exact solutions are provided with Cardano’s formula alone, it is often overlooked in classrooms because of its derivation complexity and the use of complex numbers it generates with their rather abstract 120 deg respective rotations in the polar plane to find roots 2 and 3. Indeed Cardano’s formula was originally limited to 2 real roots until Bombelli discovered complex numbers soon after.

But as it turned out, complex numbers can be avoided in application since their imaginary parts cancel out.

Of course for roots 2 and 3 we can always use the Standard Quadratic Equation, but that requires dividing the Cubic function by the known root’s factor to derive the quotient polynomial for uploading into that equation. This division step is avoided by using the Extended Quadratic Equation I presented in Cubic Polynomials-A Simpler Approach where the known factor is loaded directly into…