Solving cubic equations by completing the cube and higher degree equations by completing powers

TL;DR

This paper introduces a method for solving cubic and certain higher-degree algebraic equations by completing powers, simplifying the process to solving linear and quadratic equations, based on Harrison's center theory.

Contribution

It derives the Cardano formula through completing the cube and extends the approach to higher-degree equations using completing powers, offering a new perspective and simplified solutions.

Findings

Derived the Cardano formula via completing the cube.

Provided radical solutions for some higher-degree equations.

Presented a simple criterion for solving algebraic equations by completing powers.

Abstract

We derive the Cardano formula of cubic equations by completing the cube, and provide radical solutions to some algebraic equations of higher degree by completing powers. The main idea of completing powers arises from Harrison's center theory of higher degree forms. A very simple criterion for such algebraic equations is presented, and the computation amounts to solving linear equations and quadratic equations.