<s>[INST] Proof that expression is positive if conditions is met Is there a way to show that this expression is always positive as long as $b>0$ and $r>x$ ? Assume $r>0$ and $x>0$
\begin{equation}
b-1+\sqrt{(1+b)^2-\frac{4rb}{x}}
\end{equation}
It's simple with $b>1$ but I can't figure it out if it is true for lower values. 
 [/INST] This is false.  Consider $b = \frac{1}{10}, x = 1, r = \frac32$.  Clearly, we have $b,x,r>0$ and $r>x$.
$$\sqrt{(1+b)^2-\frac{4rb}{x}} = \sqrt{\left(1+\frac{1}{10}\right)^2-\frac{4(3/2)(1/10)}{1}} = \sqrt\frac{121-60}{100} = \frac{\sqrt{61}}{10}$$
$$b-1+\sqrt{(1+b)^2-\frac{4rb}{x}} = -\frac9{10} + \frac{\sqrt{61}}{10} = \frac{\sqrt{61}-\sqrt{81}}{10} < 0$$
</s>