Question

If x + 1/x = 5, find the values of (x² + 1/x²) and (x⁴ + 1/ x⁴).​

Answer 1

$\begin{gathered}\Large{\bold{\pink{\underline{Formula \: Used \::}}}} \end{gathered}$

$(1). \: \boxed{ \pink{\tt \: {(x + y)}^{2} = {x}^{2} + 2xy + {y}^{2}}}$

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

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

On squaring both sides we get,

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

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

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

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

On squaring both sides, we get

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

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

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

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