For polynomials of degree two that 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 symmetric vector solutions. Circular, generalized circular and hyperbolic solutions are considered. As an application, we describe the way from considering complex signals to considering vector signals. 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 usually used there.
Wolf-Dieter Richter (Sat,) studied this question.