Hmm...I can't see what the significance of the the first quadrant point is...
The $\displaystyle x$ coordinate is the solution of the equation $\displaystyle f(x)=x$ which may be expanded:
$\displaystyle \sqrt{1996x}=x^2$
and again:
$\displaystyle 1996x=x^4$
(but now we need to remember that $\displaystyle x\ge 0$) and rearranging we have:
$\displaystyle x^4-1996x=0$
which by Descartes rule of signs has exactly one positive root. So the $\displaystyle x$ coordinate of the point of intersection is the unique positive root of the quartic:
$\displaystyle x^4-1996x=0$
CB