Question

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Answer 1

Answer:

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Answer 2

$x = \frac{2}{4 + 5 \sqrt{2} } \\ y = \frac{3}{4 - 5 \sqrt{2} }$

$x + y = \frac{2}{4 + 5 \sqrt{2} } + \frac{3}{4 - 5 \sqrt{2} } \\ = \frac{2(4 - 5 \sqrt{2}) + 3(4 + 5 \sqrt{2}) }{(4 + 5 \sqrt{2} )(4 - 5 \sqrt{2}) } \\ = \frac{8 - 10 \sqrt{2} + 12 + 15 \sqrt{2} }{ {(4)}^{2} - {(5 \sqrt{2}) }^{2} } \\ = \frac{8 + 12 + 5 \sqrt{2} }{16 - 50} \\ = \frac{20 + 5 \sqrt{2} }{ - 34} \\ = \frac{ - 20 - 5 \sqrt{2} }{34} \\ answer = \frac{ - 20 - 5 \sqrt{2} }{34}$

$x - y = \frac{2}{4 + 5 \sqrt{2} } - \frac{3}{4 - 5 \sqrt{2} } \\ = \frac{2(4 - 5 \sqrt{2} ) - 3(4 + 5 \sqrt{2}) }{ {4}^{2} - {(5 \sqrt{2} )}^{2} } \\ = \frac{8 - 12 - 10 \sqrt{2} - 15 \sqrt{2} }{ - 34} \\ = \frac{ - 4 - 25 \sqrt{2} }{ - 34} = \frac{ - ( - 4 - 25 \sqrt{2} )}{34} \\ = \frac{4 + 25 \sqrt{2} }{34} \\ answer = \frac{4 + 25 \sqrt{2} }{34}$

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