Algebraic Manipulation involving square roots I have the following expressions:
$$E(g) :=  \frac{Ug^2 - 4tg}{1+g^2}  \hspace{2cm} (1)$$
$$ g_0 := \frac{1}{4t} \left[ -U + \sqrt{U^2 + (4t)^2} \right] \hspace{2cm} (2) $$
By plugging (2) in (1) I should arrive at:
$$ E(g_0) = \frac{1}{2} \left( U - \sqrt{U^2 + (4t)^2} \right) \hspace{2cm} (3)$$
But I can't get there. Here is my attempt so far:
$$ 1 + g_0^2 = 1+ \frac{1}{16t^2} \left( U^2+U^2+(4t)^2 - 2U \sqrt{ U^2 + (4t)^2} \right)$$
$$ = 1+ \frac{2U^2}{16t^2} + 1 - \frac{2U}{16t^2} \sqrt{ U^2 + (4t)^2}$$
$$ = 2 + \frac{2U}{16t^2} \left( U - \sqrt{ U^2 + (4t)^2} \right)$$
$$ -4t g_0 + Ug_0^2 = U - \sqrt{ U^2 + (4t)^2} + U \left( \frac{2U^2}{16t^2} + 1- \frac{2U}{16t^2} \sqrt{ U^2 + (4t)^2} \right)$$
$$ = - \sqrt{ U^2 + (4t)^2} + U \left( \frac{2U^2}{16t^2} + 2 - \frac{2U}{16t^2} \sqrt{ U^2 + (4t)^2}  \right)$$
$$ = - \sqrt{ U^2 + (4t)^2} + U \left(  2 + \frac{2U}{16t^2} \left\{ U - \sqrt{ U^2 + (4t)^2}  \right\} \right)$$
So:
$$ E(g_0) = \frac{- \sqrt{ U^2 + (4t)^2} + U \left(  2 + \frac{2U}{16t^2} \left\{ U - \sqrt{ U^2 + (4t)^2}  \right\} \right)}{2 + \frac{2U}{16t^2} \left( U - \sqrt{ U^2 + (4t)^2} \right)}$$
$$ = U - \frac{ \sqrt{ U^2 + (4t)^2}}{2 + \frac{2U}{16t^2} \left( U - \sqrt{ U^2 + (4t)^2} \right)}$$
I don't know how to simplify this any further.
For reference Im reading this: http://www.scholarpedia.org/article/Gutzwiller_wave_function , and working with equations (4), (5) and (6)

It follows from $\,(2)\,$ that $\require{cancel}\,(4 t g_0+U)^2 = U^2 + 16t^2\,$, or $\,16t^2g_0^2 + 8 Utg_0 + \cancel{U^2} = \cancel{U^2} + 16t^2\,$, and assuming $\,t \ne 0\,$ it follows that $\,2tg_0^2 + Ug_0-2 t = 0 \iff \color{blue}{2tg_0^2} = -Ug_0+2t\,$. Then:
$$
\begin{align}
E(g_0) &=  \frac{Ug_0^2 - 4tg_0}{1+g_0^2} \cdot \frac{2t}{2t}
\\ &=  \frac{2tg_0^2U - 8t^2g_0}{2t+\color{blue}{2tg_0^2}}
\\ &= \frac{2tg_0(Ug_0- 4t)}{2t -Ug_0+2t}
\\ &= \frac{2tg_0\cancel{(Ug_0- 4t)}}{-\cancel{(Ug_0-4t)}}
\\ &= -2tg_0
\end{align}
$$
The latter is equivalent to equality $\,(3)\,$ which was to be proved.
