Complex Roots of Quadratics with the Floor Function

Abstract

Quadratics are generally a rather well-understood portion of mathematics, with methods of solving quadratics being presented early in one’s mathematical development. However, when floor functions, also a well-understood function in mathematics, are inserted into such quadratics, the complexity of such expressions quickly increases. In this paper we extend the work of a previous paper focusing on the behavior of quadratics with the floor function in them by extending our analysis to the complex plane. In our paper, we utilize both algebraic analysis and domain coloring to examine the roots of such functions, and the impact the coefficients of a quadratic have on the roots of such a function. Our paper finds that quadratics with a floor function inside can be solved through finding the intersection of two graphs or through the use of domain coloring. We also show that there is no upper bound on the number of complex roots a single function of the form z²+b[z]+c can have. Lastly, we examine the behavior of the roots of a quadratic with a floor function inside when the coefficients of the equation are incremented and find similarities to the behavior of a normal quadratic.