Visualizing complex roots of a quadratic equation

Question

Is there some way to visualize or plot Complex roots of a quadratic equation ( of real coefficients):

$$ y= x^2 + v x + c =0 $$

whose product is constant? .. in a complex plane?

Shown are cases together with real/ coincident /double roots.

Here $v$ is variable and the constant product $c= 4. $

Plot[{x^2 + 4, x^2 - x + 4, x^2 - 2 x + 4, x^2 - 3 x + 4, 
  x^2 - 4 x + 4, x^2 - 5 x + 4}, {x, -3, 6}, GridLines -> Automatic]

Answer 1

For $c>0$:

$$z=\frac{-v\pm\sqrt{v^2-4c}}2\text{ or }\frac{-v\pm i\sqrt{4c-v^2}}2.$$

In a $(v,x,y)$ representation, an hyperbola in the $(v,x)$ plane and an ellipse in the $(v,y)$ plane. In the $(x,y)$ plane, a circle of radius $2\sqrt c$ and the portions of the axis $x$ outside of it.

Answer 2

Call the $x$-coordinate of the vertex $a$ and the roots $a\pm bi$. From the point $(a,0)$ which is the perpendicular projection of the vertex onto the $x$-axis, draw a tangent to the parabola and drop a perpendicular from the tangent point back to the $x$-axis. The distance between the two projection points you have constructed is $b$.