Question

x=3+√8 find the value of x²+1/x²

Answer 1

x= 3+√8

⇒1/x = 1/(3+√8) * (3-√8)/(3+√8) =(3-√8)/(3²-(√8)²) = 3-√8

x+1/x = (3+√8)+(3-√8) = 6

So (x+1/x)² = 6²

⇒x² + 1/x² + 2 = 36

⇒x² + 1/x² = 34

Answer 2

$\: \: \: \: \: \: \: \: \color{green} \huge \frak{Hola! \: Mate} \\ \\ \underline{\bf{QUESTION}} : \\ \\ \rightarrow\tiny \bf{x = 3 +\sqrt{8} \: \: find \: the \: value \: of \: {x}^{2} + \frac{1}{ {x}^{2} } } \\ \\ \underline{\bf{GIVEN}} : \\ \\ \rightarrow\tiny \bf{x = 3 + \sqrt{8} } \\ \\ \underline{\bf{FIND}} : \\ \\ \rightarrow \tiny \bf{the \: value \: of \: {x}^{2} + \frac{1}{ {x}^{2} } } \\ \\ \underline{\bf{SOLUTION}} : \\ \\ \rightarrow\tiny \bf{x = 3 + \sqrt{8} \: , \: \: \: \: \: \: \: \: \frac{1}{x} = \frac{1}{3 + \sqrt{8} } } \\ \\ \rightarrow \tiny \bf{ \frac{1}{3 + \sqrt{8} } } \times \frac{3 - \sqrt{8} }{3 - \sqrt{8} } = \frac{3 - \sqrt{8} }{ {(3)}^{2} - { (\sqrt{8)} }^{2} } \\ \\ \rightarrow\tiny \bf{ \frac{3 - \sqrt{8} }{9 - 8} } = \frac{3 - \sqrt{8} }{1} = 3 - \sqrt{8} \\ \\ \rightarrow\tiny \bf{x + \frac{1}{ x} = 3 + \cancel{ \sqrt{ 8}} + 3 - \cancel{ \sqrt{8} }} \\ \\ \rightarrow \tiny \bf{x + \frac{1}{x} = 6} \\ \\ \rightarrow\bf{ on \: squaring \: them} \\ \\ \rightarrow \tiny \bf{(x + \frac{1}{x} )^{2} = (6)^{2} } \\ \\ \rightarrow \tiny \bf{ {x}^{2} + \frac{1}{ {x}^{2} } + 2 = 36} \\ \\ \rightarrow \tiny \bf{ {x}^{2} + \frac{1}{ {x}^{2} } = 36 - 2 } \\ \\ \rightarrow \tiny \bf{ {x}^{2} + \frac{1}{ {x}^{2} } = 34 } \\ \\ \color{green} \underline{\huge \mathbb{HOPE \: IT'S \: HELPFUL}}$