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.

翻译：本文描述了一种通过几何方法求解一元复变量二次方程 $x^2+c_1 x+c_2=0$ 的根 $r_1$、$r_2$，该方法利用复数平面中由已知系数 $c_1$、$c_2$ 构建的一条直线 $L$ 和一个圆周 $C$。该直线-圆周几何结构将所求根 $r_1$、$r_2$ 包含在其组成元素 $L$ 与 $C$ 的交点处。直线 $L$ 通过莫比乌斯变换映射至圆周 $C$。直线 $L$ 的位置和倾斜角可直接由系数 $c_1$、$c_2$ 计算得出，而圆周 $C$ 则通过将常数项 $c_2$ 除以直线 $L$ 上的每个点来构建。本文描述并发展了直线-圆周方法的技术细节，随后通过数值示例展示了该方法的运作过程。此处所述的一元二次直线-圆周方法可推广至高于二次的一元多项式，用于求其根的初始近似值。此外，本文还研究了连接二次直线-圆周结构中关键点的直线段的一个有趣性质。