Question

if x+1/x=(√3+√2)/(√3-√2) ,find x²+1/x²

Answer 1

$x + \frac{1}{x} = \frac{ \sqrt{3} + \sqrt{2} }{ \sqrt{3} - \sqrt{2} } \\ = \frac{ {(\sqrt{3} + \sqrt{2} )}^{2} }{( \sqrt{3} - \sqrt{2} ) (\sqrt{3} + \sqrt{2} )} \\ = \frac{3 + 2 \times \sqrt{3} \times \sqrt{2} + 2 }{3 - 2} = 5 + 2 \sqrt{6} \\ \\ {x}^{2} + \frac{1}{ {x}^{2} } = {(x + \frac{1}{x} )}^{2} - 2 \\ = {(5 + 2 \sqrt{6}) }^{2} - 2 \\ = (25 + 20 \sqrt{6} + 24) - 2 \\ = 47 + 20 \sqrt{6}$