Quadratic Equation over Associative D-Algebra

TL;DR

This paper explores quadratic equations over associative D-algebras, especially quaternions, analyzing root existence and multiplicity, and adapting classical methods like Viete's theorem and completing the square.

Contribution

It extends quadratic equation analysis to associative D-algebras, providing new insights into root structures and solution methods in quaternion algebra.

Findings

In quaternion algebra, quadratic equations can have either one root or no roots.

When a is negative real, the equation x^2=a has infinitely many roots.

The paper adapts classical algebraic methods to the context of quaternion and D-algebras.

Abstract

In this paper, I treat quadratic equation over associative $D$-algebra. In quaternion algebra $H$, the equation $x^2=a$ has either $2$ roots, or infinitely many roots. Since $a\in R$, $a<0$, then the equation has infinitely many roots. Otherwise, the equation has roots $x_1$, $x_2$, $x_2=-x_1$. I considered different forms of the Viete's theorem and a possibility to apply the method of completing the square. In quaternion algebra, there exists quadratic equation which either has $1$ root, or has no roots.