Question

if a = root 3 - 2root 2 root 3 + root 2 / root 3 + root 2 and b = root 3 + root 2 / root 3 - root 2 , find the value of a square + b square - 5 ab

Answer 1

$a = \frac{ \sqrt{3} - \sqrt{2} }{ \sqrt{3} + \sqrt{2} } \\ b = \frac{ \sqrt{3} + \sqrt{2} }{ \sqrt{3} - \sqrt{2} } \\ a + b = \frac{ \sqrt{3} - \sqrt{2} }{ \sqrt{3} + \sqrt{2} } + \frac{ \sqrt{3} + \sqrt{2} }{ \sqrt{3} - \sqrt{2} } \\ = \frac{( { \sqrt{3} - \sqrt{2} })^{2} + ( { \sqrt{3} + \sqrt{2} })^{2} }{( { \sqrt{3} })^{2} - ( { \sqrt{2} })^{2} } \\ = \frac{2(( { \sqrt{3} })^{2} + ( { \sqrt{2} })^{2} ) }{3 - 2} \\ = 2(3 + 2) \\ = 10 \\ ab = 1 \\ now \: \: \: \: {a}^{2} + {b}^{2} - 5ab = ( {a + b})^{2} - 7ab \\ = ( {10})^{2} - 7 \times 1 \\ = 100 - 7 \\ = 93$