Question

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Answer 1

$\huge\color{orange}\boxed{\colorbox{black}{\sf{☣ANSWER ☣}}}$

$\huge\mathfrak\blue{First\: rationalize \: 'x'}$

$x = \frac{1}{2 - \sqrt{3} } \\ \\ = \frac{1 \times (2 + \sqrt{3} )}{(2 - \sqrt{3} ) \times (2 + \sqrt{3} )} \\ \\ = \frac{2 + \sqrt{3} }{4 - 3} \\ \\ = \frac{2 + \sqrt{3} }{1} \\ \\ = 2 + \sqrt{3}$

$\huge\sf\purple{Now \: solve \: for \: } \\ \pink{x + \frac{1}{x} }$

$x + \frac{1}{x} = 2 + \sqrt{3} + \frac{1}{2 + \sqrt{3} } \\ \\ = 2 + \sqrt{3} + \frac{1 \times (2 - \sqrt{3} )}{(2 + \sqrt{3} ) \times (2 - \sqrt{3}) } \\ \\ = 2 + \sqrt{3} + \frac{2 - \sqrt{3} }{4 - 3} \\ \\ = 2 + \sqrt{3} + \frac{2 - \sqrt{3} }{1} \\ \\ = 2 + \sqrt{3} + 2 - \sqrt{3} \\ \\ = 2 + 2 \\ \\ = 4$

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