The Complex Roots of a Quadratic Equation: A Visualization

Visualization is a powerful tool in any teacher's bag of tricks, perhaps even more so for the mathematics teacher who must often visually present abstract concepts. At times, though, we may be at a loss as to how to visually represent a particular notion. For example, when the roots of a quadratic equation are real numbers, the graph of the corresponding quadratic function intersects the horizontal axis in two points (or just one point if the root is of multiplicity 2). See Figure 1. In this case, the student can easily see the relationship between the roots and the graph of the function along with other concepts such as the symmetry of the roots about the axis of the parabola. However when the roots are complex the graph does not intersect the $x$-axis, as shown in Figure 2, so how do we "see" them in this case?