The Cubic Equation Made Simple

TL;DR

This paper introduces the MY function, offering an intuitive and simplified approach to solving cubic equations, bridging the gap between trigonometric and radical solutions with a smooth transition and accurate approximations.

Contribution

The paper presents the MY function as a novel tool for solving cubic equations, providing a new perspective and an accurate algebraic approximation that simplifies traditional methods.

Findings

MY function offers a smooth transition between different solution forms.

An accurate closed-form algebraic approximation of MY was developed.

MY behaves like power functions but cannot be expressed in real radicals.

Abstract

This article introduces an intuitive function MY that simplifies solving cubic equations without venturing into the complex space. To many, it's quite strange that cubic root(s) are expressed using trigonometric functions in the three-real-roots case versus real-radicals in the one-real-root case. Yet, the MY function provides a different perspective to this oddity and shows that the transition between the two worlds is actually smooth. Although MY's behavior resembles power functions, it cannot be expressed in real radicals. That said, we succeeded in proving an accurate closed form algebraic approximation of MY. So yes, casus irreducibilis still holds, but real radicals can get you very close!