Visualizing the real and complex roots of .
When the graph of intersects the x-axis, the roots are real and we can visualize them on the graph as x-intercepts.
But what about when there are no real roots, i.e. when the graph does not intersect the x-axis? The equation still has 2 roots, but now they are complex. This visual imagines the cartesian graph floating above the real (or x-axis) of the complex plane. In this manner, real roots correspond with traditional x-intercepts, but now we can see some of the symmetry in how the complex roots relate to the original graph.
(Note: Despite showing complex roots, this plot makes sense only for the real-valued function . In other words, only for input values, x, that result in real values of .)

Question: Why are the complex roots always the points where grey curve hits the complex plane?