Visualizing Complex Roots

This comes via Glenn Waddell:

But hold on, see those 2 points where it crosses the X-axis? And see the Axis of Symmetry that goes through both equations? If we use those three points as definitions for a circle, we get the following graph and equation.

graph5 (x-2)^2 + y^2 = 2

Guess what the solution to the quadratic equation y = x^2 –4x + 6 is. If you guessed 2 + root(2)i and 2 – root(2)i  then you are absolutely correct.

The real number part of the complex solution of a quadratic with two imaginary roots is the X value of the Axis of Symmetry, and the imaginary part of the solution is the radius of the circle created by the center and endpoints created when the inverted parabola crosses the X-Axis!

Here’s another site that describes this same technique for visualizing the solutions. Also, James Tanton writes about this method in his “Guide to Everything Quadratic” (page 53).

I’m not sure how to turn this into an insightful activity for students, but I think that’s probably because I haven’t given it enough thought. If you have something, please share it.

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