Question

if x=9+4√5 then find= (√x-1/√x)^2

Answer 1

$x = 9 + 4 \sqrt{5} \\ x = 5 + 4 + 4 \sqrt{5} \\ x = { (\sqrt{5}) }^{2} + {(2)}^{2} + 2 \times 2 \times \sqrt{5} \\ x = {( \sqrt{5} + 2)}^{2} \\ \sqrt{x} = \sqrt{5} + 2$
${ (\frac{ \sqrt{x} - 1}{ \sqrt{x} }) }^{2} \\ = {( \frac{ \sqrt{5} + 2 - 1 }{ \sqrt{5} + 2} )}^{2} \\ = { (\frac{ \sqrt{5} + 1}{ \sqrt{5} + 2 } })^{2}$

Now rationalise this to get
$( {5 + 3 \sqrt{5} + 2 })^{2} \\ = {(7 + 3 \sqrt{5} )}^{2} \\ = 94 + 42 \sqrt{5}$