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

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.

q

z=x+iy

a+q=0

2

(z-p)

x,0,a+q

2

(x-p)

x

y

q

p,y,a-+q

2

y

x

y

f(x,y)=Re+q

2

a(x+iy-p)

x

y