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.