Can you graph equations with a negative discriminant? And how do you plot complex numbers both on a 2D complex plane and a 4D complex plane?

Question

I don't understand the relationship between complex numbers and that way they are graphed. The equation I am working with is $2x^{2} - 6x + 5 = 0$ where my two roots are complex solutions: $x = (3/2)-(i/2)$ and $x = (3/2)+(i/2)$. From what I understand, there is different dimensional space where these roots can be plotted due to the fundamental theorem of algebra. Although I don't quite fully understand the fundamental theorem of algebra apart from that a complex polynomial of degree $n$ has precisely $n$ roots.

Answer 1

A different color is the usual way. You plot everything multiplied by $i$ in a different color than the real part.

Try plotting Euler's identity in mathematica; Plot[E^(I x), {x, -6 Pi, 6 Pi}] which is $$y=\cos(x)+i \sin(x)$$ to get a better understanding of what I'm saying.

To your original question, the roots of a polynomial with a negative discriminant can be marked as different colored points in the real plane or be put in a complex plane. For the 4D part, you can 3D plot a colored contour similar to the way you plot a 2D representation of a 3D object by using contour plots but reading that to get information out of it might require a lot of work.