Question

If x = 9+4√5 then x = √x - 1/√x

Answer 1

Question :–

• If x = 9+4√5 then √x - 1/√x = ?

ANSWER :–

GIVEN :–

• x = 9 + 4√5

TO FIND :–

• √x - 1/√x = ?

SOLUTION :–

• First we have to find 1/x –

$\\ \implies { \bold{ \dfrac{1}{ x } = \dfrac{1}{9 + 4 \sqrt{5} } }}$

• Now rationalization –

$\\ \implies { \bold{ \dfrac{1}{ x } = \dfrac{1}{9 + 4 \sqrt{5} } \times \dfrac{9 - 4 \sqrt{5} }{9 - 4 \sqrt{5} } }}$

$\\ \implies { \bold{ \dfrac{1}{ x } = \dfrac{9 - 4 \sqrt{5} }{(9 - 4 \sqrt{5})(9 + 4 \sqrt{5})} }}$

$\\ \implies { \bold{ \dfrac{1}{ x } = \dfrac{9 - 4 \sqrt{5} }{(9) {}^{2} - (4 \sqrt{5}) {}^{2}} \: \: \: \: [ \because(a + b)(a - b) = {a}^{2} - {b}^{2} ]}}$

$\\ \implies { \bold{ \dfrac{1}{ x } = \dfrac{9 - 4 \sqrt{5} }{81 - 80} }}$

$\\ \implies { \bold{ \dfrac{1}{ x } = {9 - 4 \sqrt{5}} }}$

• Now let's find –

$\\ { \bold{ = \sqrt{x} - \dfrac{1}{ \sqrt{x} } }}$

• We should write this as –

$\\ { \bold{ = \sqrt{ ( \sqrt{x} - \dfrac{1}{ \sqrt{x} }) {}^{2} }}}$

$\\ { \bold{ = \sqrt{ x + \dfrac{1}{x} - 2( \sqrt{x} )( \dfrac{1}{ \sqrt{x} }) }}}$

$\\ { \bold{ = \sqrt{ 9 + 4 \sqrt{5} + 9 - 4 \sqrt{5} - 2 }}}$

$\\ { \bold{ = \sqrt{ 18 - 2 }}}$

$\\ { \bold{ = \sqrt{ 16 }}}$

$\\ { \bold{ = \pm \: 4 }}$

• Hence , $\: \: \: \large { \boxed{ \bold{ \sqrt{x} - \dfrac{1}{ \sqrt{x} } = \pm \: 4 }}}$