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.
On
Since you say "not on an Argand diagram" I suspect you are asking about whether or not the complex solutions can be seen on a Cartesian graph of the function $y= 2x^2+ 3x+ 23$. Yes, but it is subtle!
Complete the square: $2x^2+ 3x+ 23= 2(x^2+ (3/2)x)+ 23= 2(x^2+ (3/2)x+ 9/16- 9/16)+ 23= 2(x+ 3/4)^2- 9/8+ 23= 2(x+ 3/4)^2+ 193/8$. Solving for x we get $x= -3/4\pm\frac{\sqrt{193}}{4}$.
And from $y= 2(x+ 3/4)+ 193/8$ we can see that the graph is a parabola with vertex at (-3/4, 193/8). The x coordinate of the vertex is the real part of the solution. The imaginary part is a little subtler. It is the square root of the of the y coordinate divided by the leading coefficient.
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
$$(u,-v).$$