The Parabola's Evil Twin: Real and Nonreal Roots of a Real Quadratic
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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.
Contributed by: Phil Ramsden (December 2012)
Open content licensed under CC BY-NC-SA
Snapshots
Details
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?
Permanent Citation
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