Quadratic unilateral polynomials over split quaternions

TL;DR

This paper develops explicit formulas for solving quadratic equations over split quaternions, especially when the leading coefficient is non-invertible, extending previous methods to more general cases.

Contribution

It provides new explicit root formulas for quadratic split quaternion polynomials with non-invertible leading coefficients, expanding existing solution techniques.

Findings

Derived explicit root formulas for quadratic split quaternion polynomials

Verified results using methods adapted from prior work on companion polynomials

Extended solution methods to cases with non-invertible leading coefficients

Abstract

In this paper, we derive explicit formulas for computing the roots of $ax^{2}+bx+c=0$ with $a$ being not invertible in split quaternion algebra. We also imitate the approach developed by Opfer, Janovska and Falcao etc. to verify our results when the corresponding companion polynomials are not identically vanishing.