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

This paper describes a geometrical method for finding the roots of a quadratic equation in one complex variable, by means of a Line L and a Circumference C in the complex plane, constructed from the known coefficients of the quadratic equation. This Line-Circumference (LC) geometric structure contains the sought roots at the intersections of its component elements L and C. Line L is mapped onto Circumference C by a Möbius transformation. The location and inclination angle of L can be computed directly from the coefficients, while C is constructed by dividing the constant term of the quadratic equation 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 can be extended to find initial approximations to the roots of polynomials in one variable of degree greater than two. As an additional feature, this paper also studies an interesting property of the rectilinear segments connecting key points in a quadratic LC structure.