Question

if x = 9 - 4√5 find the value of √x – 1/√x

Answer 1

$given \: \: x = 9 - 4 \sqrt{5} \\ so \: \: \frac{1}{x} = \frac{1}{9 - 4 \sqrt{5}} = \frac{9 + 4 \sqrt{5}}{(9 - 4 \sqrt{5})(9 + 4 \sqrt{5})} \\ = \frac{9 + 4 \sqrt{5}}{81 - 80} = 9 + 4 \sqrt{5} \\ \\ therefore \: \: x + \frac{1}{x} \\ = (9 - 4 \sqrt{5}) + (9 + 4 \sqrt{5}) = 18 \\ \\ x + \frac{1}{x} = {(\sqrt{x} - \frac{1}{ \sqrt{x} } )}^{2} + 2 \\ 18 = {(\sqrt{x} - \frac{1}{ \sqrt{x} } )}^{2} + 2 \\ {(\sqrt{x} - \frac{1}{ \sqrt{x} } )}^{2} = 18 - 2 = 16 \\ \sqrt{x} - \frac{1}{ \sqrt{x} } = \sqrt{16} = 4 \: \: or \: \: - 4$