Question

If $x=9-4\sqrt{5}$ then find the value of $\sqrt{x}+\frac{1}{\sqrt{x}}$.

Answer 1

$x=9-4\sqrt{5}$

$x=(5+4-2(2\left)\sqrt{5}\right)$

$x=(\sqrt{5}-2{)}^2$

$\sqrt{x}=\sqrt{5}-2$

Then $\sqrt{x}+\frac{1}{\sqrt{x}}$$=\sqrt{5}-2+\frac{1}{\sqrt{5}-2}$

$=\sqrt{5}-2+\sqrt{5}+2$

$\sqrt{x}+\frac{1}{\sqrt{x}}=2\sqrt{5}$.