What is the graphical representation of complex roots on the real Cartesian plane? Consider the quadratic $y=2x^2+3x+23$, this has no real solutions, so doesn't intercept the $x$-axis, but still has a complex conjugate pair as solutions.

My question is: Do these complex solutions have any meaning graphically on an $(x,y)$ coordinate axis (not an argand diagram)? 

I have heard that a Reimann surface may have something to do with this, but I am not too sure what that is, thanks.

If $z_0=u+iv$ and $ \overline{z_0}=u-iv$ are the solutions of the equation $ 2x^2+3x+23=0$, then the graphical representations on the real Cartesian plane are
$$(u,v)$$
and 
$$(u,-v).$$
