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 $r_1$, $r_2$ 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 $L$ in the LC structure is mapped onto Circumference $C$ by a Mobius transformation. The location and inclination angle of Line $L$ can be computed directly from coefficients $c_1$, $c_2$, while Circumference $C$ is constructed by dividing the constant term $c_2$ by each point from Line $L$. This paper describes and develops the technical details for the LC Method, and then shows how the LC Method works through a numerical example. The quadratic LC method described here can be extended to polynomials in one variable of degree greater than two, in order to find initial approximations to their roots. As an additional feature, this paper also studies an interesting property of the rectilinear segments connecting key points in a quadratic LC structure.