# 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(z-p)+q=0

2

x,0,a(x-p)+q

2

x

y

q

(p,y,a(-y+q))

2

x

y

f(x,y)=Rea(x+iy-p)+q

2

x

y