If $x=5+2 \sqrt{6}$, then $\frac{x-1}{\sqrt{x}}$ is equal to :

$x=5+2 \sqrt{6}$

$=5+2 \times \sqrt{3} \times \sqrt{2}$

$=3+2+2 \times \sqrt{3} \times \sqrt{2}$

$=(\sqrt{3}+\sqrt{2})^{2}$

$\therefore \quad \sqrt{x}=\sqrt{3}+\sqrt{2}$

$\therefore \quad \frac{1}{\sqrt{x}}=\frac{1}{\sqrt{3}+\sqrt{2}}$

$=\frac{\sqrt{3}-\sqrt{2}}{(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})}=\sqrt{3}-\sqrt{2}$

$\therefore \frac{x-1}{\sqrt{x}}$

$=(5+2 \sqrt{6}-1)(\sqrt{3}-\sqrt{2})$

$=(4+2 \sqrt{6})(\sqrt{3}-\sqrt{2})$

$=4 \sqrt{3}+2 \sqrt{18}-4 \sqrt{2}-2 \sqrt{12}$

$=4 \sqrt{3}+6 \sqrt{2}-4 \sqrt{2}-4 \sqrt{3}$

$=2 \sqrt{2}$