Question

if x=
$9 - 4 \sqrt{5}$
Find
$\sqrt{x} - 1 \div \sqrt{x}$
​

Answer 1

$\textbf {\underline{\underline{ANSWER}}}$

$a= 9 - 4 \sqrt{5} \\ \\ \\ \sqrt{a} = { \sqrt{9 - 4 \sqrt{5} } } \\ \\ \\ \sqrt{9 - 4 \sqrt{5} } = \sqrt{x} - \sqrt{y} \\ \\ \\ squaring \: both \: sides \\ \\ \\ 9 - 4 \sqrt{5} = x + y - 2 \sqrt{xy} \\ \\ \\ x + y = 9 \\ \\ \\ - 2 \sqrt{xy} = - 4 \sqrt{5} \\ \\ \\ xy = 20 \\ \\ \\ solving \: this \: we \: get \: \\ \\ \\ x = 4 \: \: \: \: \: \: \: \: \: \: \: y = 5 \\ \\ \\ so \: \\ \\ \\ \sqrt{9 - 4 \sqrt{5} } = \sqrt{4} - \sqrt{5} \\ \\ \\ = 2 - \sqrt{5} \\ \\ \\ \sqrt{a} - \frac{1}{ \sqrt{a} } = 2 - \sqrt{5} - \frac{1}{ \sqrt{2 - \sqrt{5} } } \\ \\ \\ = 2 - \sqrt{5} - \frac{ 2 + \sqrt{5} }{(2 - \sqrt{5})(2 + \sqrt{5)} } \\ \\ \\ = 2 - \sqrt{5} - \frac{2 + \sqrt{5} }{ - 1} \\ \\ \\ = 2 - \sqrt{5} + 2 - \sqrt{5} \\ \\ \\ = 4$

THANKS