Figure 1

Everyone learns that the roots of a polynomial have a
graphical interpretation: they're the places where the
function crosses the xaxis. But what happens when the
equation has only imaginary roots? Do those have a
graphical interpretation as well?
Here's an interpretation that works for quadratics.
Take a quadratic, such as 2x^{2}  8x + 10, and
graph it. In the Figure 1, it is shown in
red. Because it lies entirely
above the xaxis, we know it has no real roots.
Now, reflect the graph of this quadratic
through its bottommost point,
and find the xintercepts of this new graph,
shown in green. Finally, treat these intercepts as if
they were on opposite sides of a perfect circle, and
rotate them both exactly 90 degrees. These new points
are shown in blue.
If interpreted as points in the complex plane,
the blue points are exactly the roots of the original
equation! (In our example, they are 2+i and 2i.)
Presentation Suggestions:
Use different colors of chalk to represent various parts
of the diagram.
The Math Behind the Fact:
You can prove it by expanding the generic equations:
(xa)^{2} + b and (xa)^{2} + b
and then comparing their roots
(using the quadratic equation).
How to Cite this Page:
Su, Francis E., et al. "Complex Roots Made Visible."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
