Question

if (X ^2 + 1)/X = 3 whole 1/3
and X > 1 then find:
(i)x-1/x
(ii)x^3-1/x​

Answer 1

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$\tt \frac{ {x}^{2} + 1 }{x} = 3 \frac{1}{3}$

$\tt x + \frac{1}{x} = \frac{10}{3}$

Squaring both sides,

$\tt (x + \frac{1}{x})^{2} = ( \frac{10}{3} )^{2}$

$\tt {x}^{2} + ( { \frac{1}{x} })^{2} + 2(x)( \frac{1}{x} ) = \frac{100}{9}$

$\tt {x}^{2} + ( { \frac{1}{x} })^{2} = \frac{100}{9} - 2$

$\tt {x}^{2} + ( \frac{1}{x} )^{2} = \frac{82}{9}$

Now,

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

$\tt = ( \frac{82}{9} - 2)$

$\tt = ( \frac{64}{9} )$

$\tt (x - \frac{1}{x} ) = \sqrt{ (\frac{64}{9} )} = ( \frac{8}{3}) = 2 \frac{2}{3}$

Now,

$\tt ( {x}^{3} - \frac{1}{ {x}^{3} } ) = (x - \frac{1}{x} ) + ( {x}^{2} + \frac{1}{x} )^{2} +(x)( \frac{1}{x} )$

$\tt = ( \frac{8}{3} )( \frac{82}{9} + 1)$

$\tt (\frac{8}{3} )( \frac{91}{9} ) = \frac{728}{27} = 26 \frac{26}{7}$

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