Question

plz ans this question​

Answer 1

$x = \frac{ \sqrt{5} - \sqrt{2} }{ \sqrt{5} + \sqrt{2} }$

Rationalise The value of X

$\frac{ \sqrt{5} - \sqrt{2} }{ \sqrt{5} + \sqrt{2} } \times \frac{ \sqrt{5} - \sqrt{2} }{ \sqrt{5} - \sqrt{2} } \\ \\ \frac{ (\sqrt{5} - \sqrt{2}) ^{2} }{ { \sqrt{5} }^{2} - { \sqrt{2} }^{2} } \\ \\ \frac{ { \sqrt{5} }^{2} + \sqrt{2} {}^{2} - 2 \times \sqrt{5} \times \sqrt{2} }{5 - 2} \\ \\ \frac{5 + 2 - 2 \sqrt{10} }{3} \\ \\ \frac{7 - 2 \sqrt{10} }{3}$

$x = \frac{7 - 2 \sqrt{10} }{3}$

$y = \frac{ \sqrt{5} + \sqrt{2} }{ \sqrt{5} - \sqrt{2} }$

Rationalise the Value of Y

$\frac{ \sqrt{5} + \sqrt{2} }{ \sqrt{5} - \sqrt{2} } \times \frac{ \sqrt{5} + \sqrt{2} }{ \sqrt{5} + \sqrt{2} } \\ \\ \frac{( \sqrt{5} + \sqrt{2} ) ^{2} }{ { \sqrt{5} }^{2} - \sqrt{2} {}^{2} } \\ \\ \frac{ \sqrt{5} {}^{2} + \sqrt{2} {}^{2} + 2 \times \sqrt{5} \times \sqrt{2} }{5 - 2} \\ \\ \frac{5 + 2 + 2 \sqrt{10} }{3} \\ \\ \frac{7 + 2 \sqrt{10} }{3}$

$y = \frac{7 + 2 \sqrt{10} }{3}$

We have to Find

${x}^{2} + xy + {y}^{2} \\ \\ ( \frac{7 - 2 \sqrt{10} }{3} ) ^{2} + ( \frac{7 - 2 \sqrt{10} }{3} )( \frac{7 + 2 \sqrt{10} }{3} ) + ( \frac{7 + 2 \sqrt{10} }{3} ) {}^{2} \\ \\ ( \frac{49 + 40 - 28 \sqrt{10} }{9} ) + (\frac{7 }{3}) {}^{2} - (\frac{2 \sqrt{10} }{3} ) {}^{2} + \frac{49 + 40 + 28 \sqrt{10} }{9} \\ \\ \frac{89 - 28 \sqrt{10} }{9} + \frac{9}{9} + \frac{89 + 28 \sqrt{10} }{9} \\ \\ \frac{89 - 28 \sqrt{10} + 9 + 89 + 28 \sqrt{10} }{9} \\ \\ \frac{89 + 9 + 89}{9} \\ \\ \frac{187}{9}$

$\boxed{\mathbf{\huge{Answer=\frac{187}{9}}}}$