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.
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?