The Parabola's Evil Twin: Real and Nonreal Roots of a Real Quadratic

For negative , the roots of the quadratic equation are found where the parametric curve (the blue parabola) intersects the - plane. However, for positive , they are found where (the red "evil twin") intersects the - plane. The blue and red parabolas are the intersections of the surface with the two vertical planes through its saddle point, parallel to the and axes, respectively.


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The idea for this neat visualization of real and complex-conjugate roots of quadratic equations is due to David Wilson (personal communication).
The most important control is the slider. The default value of is negative; real roots are shown in the - plane as blue points. Increase the value of and the red "evil twin" takes over; the roots become nonreal and are shown as red points.
Sliders are also provided for the parameters and ; it may be instructive to try to predict their effect. What happens when becomes negative?
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