A Geometric Interpretation of Complex Zeros of Quadratic Functions

Most high school mathematics students learn how to determine the zeros of quadratic functions such as f (x) = ax^2 + bx + c, where a, b, and c are real numbers. At some point, students encounter a quadratic function of this form whose zeros are imaginary or complex-valued. Since the graph of such functions do not intersect the x-axis in the xy-plane, students may be left with the impression that complex-valued zeros of quadratics cannot be visualized. The main purpose of this manuscript is to show that if the zeros of a quadratic function with real-valued coefficients are imaginary, the zeros can be seen if we use an appropriate coordinate system. For illustrative purposes, we have used the software program GeoGebra, which allows us to create a three-dimensional Cartesian coordinate system where imaginary zeros can be viewed simultaneously with the graph of the quadratic function they correspond to. To illustrate this, we will apply geometric transformations to the function given by f (x) = x^2 − 6x + 13 in order to visualize its zeros, which happen to be complex-valued. Then, we will identify a particular set of complex numbers that can be used as inputs for the function f. Using this set of complex numbers, we can construct the exact image that is produced by the geometric transformations. Then, we may deem the two methods as equivalent ways to ultimately construct the geometric images of complex-valued zeros of quadratic functions with real-valued coefficients.Â