Question

If
$\frac{2 + \sqrt{5} }{2 - \sqrt{5} } = x$
and
$\frac{2 - \sqrt{5} }{2 + \sqrt{5} } = \: y$

find the value of
${x}^{2} - {y}^{2}$

​

Answer 1

$x = \frac{2 + \sqrt{5} }{2 - \sqrt{5} } = \frac{(2 + \sqrt{5})(2 + \sqrt{5} ) }{(2 - \sqrt{5})(2 + \sqrt{5}} = \frac{4 + 5 + 4 \sqrt{5} }{4 - 5} = - 9 - 4 \sqrt{5} \\ y = \frac{2 - \sqrt{5} }{2 + \sqrt{5} } =\frac{(2 - \sqrt{5})(2 - \sqrt{5} ) }{(2 + \sqrt{5})(2 - \sqrt{5}} = 4 \sqrt{5} - 9\\ {x}^{2} - {y}^{2} \\ = (x + y)(x - y) \\ = ( - 9 - 4 \sqrt{5} + 4 \sqrt{5} - 9)( - 9 - 4 \sqrt{5} - 4 \sqrt{5} + 9) \\ = - 18 \times ( - 8 \sqrt{5} ) \\ = 144 \sqrt{5}$