Question

please help me to find the answer

Answer 1

$We \: Have \: = > x = \frac{5 - \sqrt{21} }{2} \\ \\ Then \: Find \: \\ \\ ( {x}^{3} + \frac{1}{{x}^{3} } ) - 5( {x}^{2} + \frac{1}{ {x}^{2} } ) + (x + \frac{1}{x} ) \\ \\ = > ( {x}^{3} - 5 {x}^{2} + x) + ( \frac{1}{ {x}^{3} } - \frac{5}{ {x}^{2} } + \frac{1}{x} ) \\ \\ = > ( {x}^{3} - 5 {x}^{2} + x) + ( \frac{1 - 5x + {x}^{2} }{ {x}^{3} } ) \\ \\ = > \frac{ {x}^{6} - 5 {x}^{5} + {x}^{4} + 1 - 5x + {x}^{2} }{ {x}^{3} } \\ \\ = > \frac{ {x}^{4}( {x}^{2} - 5x + 1) + 1( {x}^{2} - 5x + 1) }{ {x}^{3} } \\ \\ = > \frac{( {x}^{2} - 5x + 1)( {x}^{4 } + 1)}{ {x}^{3} } \: \: \: .............Eq _{1} \\ \\ Now, \: We \: Take \: term \: \\ \\ = > {x}^{2} - 5x + 1 \: \: \: \: ........Eq _{2} \\ \\ and \: as \: given \: = > x = \frac{5 - \sqrt{21} }{2} \\ \\ We \: s Squaring \: Both \: Side \: and \: We \: get \\ \\ = > {x}^{2} = {( \frac{5 - \sqrt{21} }{2} \: ) }^{2} \\ \\ = > {x}^{2} = \frac{25 - 10 \sqrt{21} + 21}{4} \\ \\ = > {x}^{2} = \frac{46 - 10 \sqrt{21} }{4} = \: ( \frac{23}{2} - \frac{5 \sqrt{21} }{2} ) \\ \\ Now ,\: Ee \: Substitute \: all \: the \: value \: in \: Eq _{2} \: We \: get \\ \\ = > ( \frac{23}{2} - \frac{5 \sqrt{21} }{2} ) - 5( \frac{5 - \sqrt{21} }{2} ) + 1 \\ \\ = > \frac{23}{2} - \frac{5 \sqrt{21} }{2} - \frac{25}{2} + \frac{5 \sqrt{21} }{2} + 1 \\ \\ = > \frac{23}{2} - \frac{25}{2} + 1 \\ \\ = > \frac{23 - 25}{2} + 1 \\ \\ = > - 1 + 1 \\ \\ = > 0 \: \: \: \: \: \\ \\ so \: \\ \\ {x}^{2} - 5x + 1 = 0 \: \: \: \: Substitute \: that \: value \: in \: Eq _{1} \\ and \: get \\ \\ = > [\frac{( {x}^{4 } + 1) \times (0)}{ {x}^{3} } ] \\ \\ = > 0 \: \: \: \: \: \: \: \: \: \: ...........Answer \\ \\ \\ \\ \\ \\ BE \: \: BRAINLY$