Question

please help me with this fast​

Answer 1

Step-by-step explanation:

We have,

$\rm \: x = \frac{ \sqrt{3} - \sqrt{2} }{ \sqrt{3} + \sqrt{2} } \\ y = \frac{ \sqrt{3} + \sqrt{2} }{ \sqrt{3} - \sqrt{2} }$

Now,

$\rm \: {x}^{2} + {y}^{2} + xy$

$\rm \: = {x}^{2} + {y}^{2} + 2xy - xy$

$\rm \: = {(x + y)}^{2} - xy$

$\rm \: = { \bigg( \frac{ \sqrt{3} - \sqrt{2} }{ \sqrt{3} + \sqrt{2} } + \frac{ \sqrt{3} + \sqrt{2} }{ \sqrt{3} - \sqrt{2} } \bigg)}^{2} - \bigg( \frac{ \sqrt{3} - \sqrt{2} }{ \sqrt{3} + \sqrt{2} } \bigg). \bigg( \frac{ \sqrt{3} + \sqrt{2} }{ \sqrt{3} - \sqrt{2} } \bigg)$

$\rm \: = { \bigg \{ \frac{ (\sqrt{3} - \sqrt{2} ) ^{2} + (\sqrt{3} + \sqrt{2} ) ^{2} }{( \sqrt{3} + \sqrt{2})( \sqrt{3} - \sqrt{2}) } \bigg \}}^{2} - 1$

$\rm \: = { \bigg \{ \frac{ 2 \{(\sqrt{3} )^{2} + (\sqrt{2} ) ^{2} \} }{(\sqrt{3})^{2} - (\sqrt{2})^{2} } \bigg \}}^{2} - 1$

$\rm \: = { \bigg \{ \frac{ 2 \{3 + 2 \} }{3 - 2 } \bigg \}}^{2} - 1$

$\rm \: = { \bigg \{ \frac {2 \times 5 }{1} \bigg \}}^{2} - 1$

$\rm \: = (10)^{2} - 1$

$\rm \: = 100 - 1 = 99$