This equation has roots z₁ and z₂ given by

-p ±   ______ Öp2 - 4q   2

so

z1 = -p +   ______ Öp2 - 4q   2 = u + iv

and

z2 = -p -   ______ Öp2 - 4q   2 = u - iv

where u+iv and u-iv are complex conjugates.

The product of the complex conjugate roots is given by

z1z2 = (u+iv)(u-iv) = u2 + v2 = |z1|2 = |z2|2.

Hence as the quadratic changes keeping q fixed and varying p the locus of the complex roots is the circle with radius Öq centre at the origin.