Complex Roots Made Visible — Math Fun Facts

This is a neat way you can visualize complex roots of quadratics. What you do is take a quadratic like:
2x^2 – 8x + 10
If you tried to factor or find real roots of this quadratic, you will run into problems. In fact, this polynomial has two complex roots (but no real roots).

According to Mudd Math Fun Facts, you now:

reflect the graph of this quadratic through its bottom-most point, and find the x-intercepts 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 2-i.)

Source:
Su, Francis E., et al. “Complex Roots Made Visible.” Math Fun Facts. http://www.math.hmc.edu/funfacts

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