Question

if x=3+√8 find the values of x+1/x and x²+1/x2​

Answer 1

$x = 3 + \sqrt{8} \\ \frac{1}{x} = \frac{1}{3 + \sqrt{8} } \\ = \frac{3 - \sqrt{8} }{9 - 8} \\ (rationalizing \: denominator) \\ \frac{1}{x} = 3 - \sqrt{8} \\ \implies \: x + \frac{1}{x} = 6 \\ ( {x + \frac{1}{x} })^{2} = 36 \\ \implies{x}^{2} + \frac{1}{ {x}^{2} } + 2 = 36 \\ \implies {x}^{2} + \frac{1}{ {x}^{2} } = 34$

Answer 2

$\huge\color{cyan}\boxed{\colorbox{black}{ANSWER ❤}}$

$x + \frac{1}{x} = 3 + \sqrt{8} + \frac{1}{3 + \sqrt{8} }$

$= 3 + \sqrt{8} + \frac{1(3 - \sqrt{8}) }{(3 + \sqrt{8} )(3 - \sqrt{8} )}$

$= 3 + \sqrt{8} + \frac{3 - \sqrt{8} }{9 - 8}$

$= 3 + \sqrt{8} + 3 - \sqrt{8}$

$= 6$

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${x}^{2} + \frac{1}{ {x}^{2} }$

$= {(x)}^{2} + {( \frac{1}{x} )}^{2} + 2(x)( \frac{1}{x} ) - 2(x)( \frac{1}{x} )$

$= {(x + \frac{1}{x}) }^{2} - 2$

$= {(6)}^{2} - 2$

$= 36 - 2$

$= 34$