Question

$\text{If}x=9+4\sqrt{5},\text{then}\sqrt{x}-\frac{1}{x}=$ _______.

• A. $1$
• B. $16$
• C. $4$
• D. $2\sqrt{5}$

Answer 1

The correct option is C

$4$

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

$x=4+5+4\sqrt{5}$

$=(2{)}^2+(\sqrt{5}{)}^2+2\left(2\right)\left(\sqrt{5}\right)$

$=(2{)}^2+2(2\left)\right(\sqrt{5})+(\sqrt{5}{)}^2$

$=(2+\sqrt{5}{)}^2$

$\therefore \sqrt{x}=2+\sqrt{5}\dots \dots \left(i\right)$

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

$=\frac{2-\sqrt{5}}{4-5}$

$=\sqrt{5}-2\dots \dots \left(ii\right)$

$\sqrt{x}-\frac{1}{\sqrt{x}}=2+\sqrt{5}-(\sqrt{5}-2)$ (from (i) and (ii))

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

$=4$