Locating the roots of a quadratic equation in one variable through a Line-Circumference (LC) geometric construction in the plane of complex numbers

Abstract

This paper describes a geometrical method for finding the roots $r_1$, $r_2$ of a quadratic equation in one complex variable of the form $x^2+c_1 x+c_2=0$, by means of a Line $L$ and a Circumference $C$ in the complex plane, constructed from known coefficients $c_1$, $c_2$. This Line-Circumference (LC) geometric structure contains the sought roots $r_1$, $r_2$ at the intersections of its component elements $L$ and $C$. Line $L$ is mapped onto Circumference $C$ by a Mobius transformation. The location and inclination angle of $L$ can be computed directly from coefficients $c_1$, $c_2$, while $C$ is constructed by dividing the constant term $c_2$ by each point from $L$. This paper describes the technical details for the quadratic LC method, and then shows how the quadratic LC method works through a numerical example. The quadratic LC method described here, although more elaborate than the traditional quadratic formula, can be extended to find initial approximations to the roots of polynomials in one variable of degree $n \geq 3$. As an additional feature, this paper also studies an interesting property of the rectilinear segments connecting key points in a quadratic LC structure.