Question

x =√5+1/√5-1 y =√5-1/√5+1 x²+xy+y²​

Answer 1

$x = \frac{ \sqrt{5} + 1 }{ \sqrt{5} - 1 } > > {x}^{2} = \frac{5 + 1 + 2 \sqrt{5} }{5 + 1 - 2 \sqrt{5} } \\ \\ y = \frac{ \sqrt{5} - 1 }{ \sqrt{5} + 1} > > {y}^{2} = \frac{5 + 1 - 2 \sqrt{5} }{5 + 1 + 2 \sqrt{5} }$

Hence,

${x}^{2} + xy + {y}^{2} \\ = \frac{6 + 2 \sqrt{5} }{6 - 2 \sqrt{5} } + (\frac{6 + 2 \sqrt{5} }{6 - 2 \sqrt{5} } \times \frac{6 - 2 \sqrt{5} }{6 + 2 \sqrt{5} } ) + \frac{6 - 2 \sqrt{5} }{6 + 2 \sqrt{5} } \\ = \frac{3 + \sqrt{5} }{3 - \sqrt{5} } + \frac{3 - \sqrt{5} }{3 + \sqrt{5} } + 1 \\ = \frac{ ({3 + \sqrt{5}) }^{2} + ({3 - \sqrt{5} )}^{2} }{(3 + \sqrt{5})(3 - \sqrt{5} )} \\ = \frac{9 + 5 + 2 \sqrt{5} + 9+ 5 - 2 \sqrt{5} }{9 - 5} \\ = \frac{18 + 10}{4} = \frac{28}{4} = 7$