Question

x=
$2 + \sqrt{5}$
find x²-1/x²


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Answer 1

$x = 2 + \sqrt{5} \\ \frac{1}{x} = \frac{1}{2 + \sqrt{5} } \\ = \frac{2 - \sqrt{5} }{4 - 5} \\ = \sqrt{5} - 2 \\ x + \frac{1}{x} = 2 + \sqrt{5} + \sqrt{5} - 2 \\ = 2 \sqrt{5} \\ x - \frac{1}{x} = 2 + \sqrt{5} - \sqrt{5} + 2 \\ = 4 \\ {x}^{2} - \frac{1}{ {x}^{2} } = (x - \frac{1}{x} )(x + \frac{1}{x} ) \\ = 4 \times 2 \sqrt{5} \\ = 8 \sqrt{5}$