Question

Find the value of: x-1/x and x²- 1/ x² if x+ 1/ x=√ 5

Answer 1

$\large\underline\blue{\bold{Solution :- }}$

$\bf\implies \:x + \dfrac{1}{x} = 5$

On squaring both sides we get,

$\rm \implies\: { \bigg( x + \dfrac{1}{x} \bigg)}^{2} = {5}^{2}$

$\rm \implies\: {x}^{2} + \dfrac{1}{ {x}^{2} } + 2 \times x \times \dfrac{1}{x} = 25$

$\rm \implies\: {x}^{2} + \dfrac{1}{ {x}^{2} } + 2 = 25$

$\rm \implies\:\boxed{ \red{\tt \: {x}^{2} + \dfrac{1}{ {x}^{2} } = 23}}$

On squaring both sides, we get

$\rm \implies\: { \bigg( {x}^{2} + \dfrac{1}{ {x}^{2} } \bigg)}^{2} = {23}^{2}$

$\rm \implies\: {x}^{4} + \dfrac{1}{ {x}^{4} } + 2 \times {x}^{2} \times \dfrac{1}{ {x}^{2} } = 529$

$\rm \implies\: {x}^{4} + \dfrac{1}{ {x}^{4} } + 2 = 529$

$\rm \implies\:\boxed{ \red{\tt \: {x}^{4} + \dfrac{1}{ {x}^{4} } = 527 }}$