Question

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Answer 1

Answer:

$x = \sqrt{ \frac{5 + 2 \sqrt{6} }{5 - 2 \sqrt{6} } } \\ = \sqrt{ \frac{5 + 2 \sqrt{6} }{5 - 2 \sqrt{6} } } \times \sqrt{ \frac{5 + 2 \sqrt{6} }{5 + 2 \sqrt{6} } } \\ = \sqrt{ \frac{(5 + 2 \sqrt{6})(5 + 2 \sqrt{6} ) }{(5 - 2 \sqrt{6})(5 + 2 \sqrt{6} ) } } \\ = \sqrt{ \frac{ {(5 + 2 \sqrt{6}) }^{2} }{ {(5)}^{2} - {(2 \sqrt{6}) }^{2} } } \\ = \sqrt{ \frac{ {(5 + 2 \sqrt{6} )}^{2} }{25 - 24} } \\ = \sqrt{ \frac{ {(5 + 2 \sqrt{6} )}^{2} }{1} } \\ = \sqrt{ {(5 + 2 \sqrt{6} )}^{2} } \\ = 5 + 2 \sqrt{6} \\ {x}^{2} {(x - 10)}^{2} = 1 \\ {(5 + 2 \sqrt{6} )}^{2} {(5 + 2 \sqrt{6} - 10)}^{2} = 1 \\( 25 + 24 + 20 \sqrt{6} ) {(2 \sqrt{6} - 10 + 5) }^{2} = 1 \\ (49 + 20 \sqrt{6} ) {(2 \sqrt{6} - 5) }^{2} = 1 \\ (49 + 20 \sqrt{6} )(24 + 25 + 20 \sqrt{6} ) = 1 \\ (49 + 20 \sqrt{6} )(49 - 20 \sqrt{6} ) = 1 \\ {(49)}^{2} - {(20 \sqrt{6}) }^{2} = 1 \\ 2401 - 2400 = 1 \\ 1 = 1 \\ LHS=RHS \\ Hence \: proved$