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

​
a
1
p
1
q
-1
For negative
q
, the roots
z=x+iy
of the quadratic equation
a
2
(z-p)
+q=0
are found where the parametric curve
x,0,a
2
(x-p)
+q
(the blue parabola) intersects the
x
-
y
plane. However, for positive
q
, they are found where
p,y,a-
2
y
+q
(the red "evil twin") intersects the
x
-
y
plane. The blue and red parabolas are the intersections of the surface
f(x,y)=Re
2
a(x+iy-p)
+q
with the two vertical planes through its saddle point, parallel to the
x
and
y
axes, respectively.
​

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
q
​
slider. The default value of
q
is negative; real roots are shown in the
x
-
y
plane as blue points. Increase the value of
q
and the red "evil twin" takes over; the roots become nonreal and are shown as red points.
Sliders are also provided for the parameters
p
and
a
; it may be instructive to try to predict their effect. What happens when
a
becomes negative?

External Links

Location of Complex Roots of a Real Quadratic

Permanent Citation

Phil Ramsden
​
​"The Parabola's Evil Twin: Real and Nonreal Roots of a Real Quadratic"​
​http://demonstrations.wolfram.com/TheParabolasEvilTwinRealAndNonrealRootsOfARealQuadratic/​
​Wolfram Demonstrations Project​
​Published: December 3, 2012