More about solutions of quadratic equations

TL;DR

This paper develops a flexible method for solving quadratic equations with no zeros by using accompanying variables and vector polynomials, introducing new solution classes and improving precision of known solutions.

Contribution

It introduces a novel, adaptable approach for quadratic solutions involving vector methods, extending to higher-degree equations and new solution classes.

Findings

Developed a method using accompanying variables for quadratic equations without zeros.

Introduced new vector solutions including circular, generalized circular, and hyperbolic types.

Enhanced the precision of known solutions by replacing the imaginary component.

Abstract

For polynomials of degree two which have no zeros, the method of accompanying variables is developed and zeros of associated vector polynomials are determined. Our flexible method uses a wide variety of possible vector-valued vector squaring methods and can be adopted to a wide variety of application situations. Known solutions are made much more precise by replacing the imaginary component and supplemented by introducing a whole class of new vector solutions. Circular, generalized circular and hyperbolic solutions are considered. Anyone who follows the approach of this work and considers equations of third or higher degree will come across further conclusions for the imaginary numbers used there.