Question

If x=5-√21/2 , find the value of x+1/x

Answer 1

$\sf given \: the \: value \: of \: x = \frac{5 - \sqrt{21} }{2} \\ \\ \sf \therefore \frac{1}{x} = \frac{2}{5 - \sqrt{21} } \\ \\ \sf on \: rationalising ,\ \\ \sf = \frac{2}{5 - \sqrt{21} } \times \frac{5 + \sqrt{21} }{5 + \sqrt{21} } \\ \\ \sf = \frac{2(5 + \sqrt{21}) }{(5 - \sqrt{21})(5 + \sqrt{21} ) } \\ \\ \sf = \frac{2(5 + \sqrt{21}) }{( {5)}^{2} - {( \sqrt{21} })^{2} } \\ \\ \sf = \frac{2(5 + \sqrt{21}) }{25 - 21} \\ \\ \sf = \frac{ \cancel2(5 + \sqrt{21} )}{ \cancel {4}^{2} } \\ \\ \sf = \frac{5 + \sqrt{21} }{2} \\ \\ \sf hence, \: value \: of \: x + \frac{1}{x } \: is \\ \\ \sf = \frac{5 - \sqrt{21} }{2} + \frac{5 + \sqrt{21} }{2} \\ \\ \sf = \frac{5 - \cancel{ \sqrt{21}} + 5 + \cancel {\sqrt{21} }}{2} \\ \\ \sf = \frac{10}{2} = \boxed5 \:$