Complex Solutions to Quadratic Inequalities and Their Modeling by Means of LATEX

Many people think that inequalities cannot be considered over the field C of complex numbers. And really, on C does not possess an order to make it an ordered field. But other orders exist and can be successfully used for getting answers to some questions which remain unanswered if restricted to the domain of real numbers. The paper notes some discordance in the behavior of the sets of solutions to quadratic equations and inequalities over the field of real numbers. These sets change when the discriminant changes until it is positive, but remain unchanged as soon as discriminant becomes negative. To overcome this oddity, it is suggested to make an exit to the complex plane with some order on it. The consequences of this action are proven, and illustrated by the dynamic model proposed in the work. The model is constructed with the help of LATEX.