Question

find a^2 + b^2
if a= (√3 - √2) / (√3 + √ 2)
b= (√3 + √2) / (√3 - √2)​

Answer 1

$a = \frac{ \sqrt{3} - \sqrt{2} }{ \sqrt{3} + \sqrt{2} } \\ \\ b = \frac{ \sqrt{3} + \sqrt{2} }{ \sqrt{3} - \sqrt{2} }$

$a = \frac{( \sqrt{3} - \sqrt{2} )( \sqrt{3} - \sqrt{2} )}{ {( \sqrt{3} )}^{2} - {( \sqrt{2}) }^{2} } \\ \\ \: \: \: = \frac{3 + 2 - 2 \sqrt{6} }{3 - 2} = \frac{5 - 2 \sqrt{6} }{1} = 5 - 2 \sqrt{6} \\ \\ \\ \\ b = \frac{( \sqrt{3} + \sqrt{2} )( \sqrt{3} + \sqrt{2} )}{ {( \sqrt{3} )}^{2} - {( \sqrt{2}) }^{2} } \\ \\ \: \: \: = \frac{3 + 2 + 2 \sqrt{6} }{3 - 2} = \frac{5 + 2 \sqrt{6} }{1} = 5 + 2 \sqrt{6}$

Now,

${a}^{2} + {b}^{2} = {(a + b)}^{2} - 2ab \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: = {(5 - 2 \sqrt{6} + 5 + 2 \sqrt{6}) }^{2} - 2(5 + 2 \sqrt{6} )(5 - 2\sqrt{6} )\\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: = {(5 + 5)}^{2} - 2[ {(5)}^{2} - {(2 \sqrt{6}) }^{2} ] \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: = {10}^{2} - 2(25 - 24)\\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: = 100 - 2 = 98$