Question

If x= √5-2/√5+2 and y= √5+2/√5-2 find (i) x² (ii) y² (iii) xy (iv) x² + y² + xy

Answer 1

Hope u like my process

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x = \frac{ \sqrt{5} - \sqrt{2} }{ \sqrt{5} + \sqrt{2} } \\ \\ y = \frac{ \sqrt{5} + \sqrt{2} }{ \sqrt{5} - \sqrt{2} } \\ \\ x + y = \frac{ \sqrt{5} - \sqrt{2} }{ \sqrt{5} + \sqrt{2} } + \frac{ \sqrt{5} + \sqrt{2} }{ \sqrt{5} - \sqrt{2} } \\ = \frac{ {( \sqrt{5} - \sqrt{2}) }^{2} + {( \sqrt{5} + \sqrt{2} ) }^{2} }{( \sqrt{5} - \sqrt{2})( \sqrt{5} + \sqrt{2} ) } \\ = \frac{5 + 2 - 2 \sqrt{10} + 5 + 2 + 2 \sqrt{10} }{ {( \sqrt{5} )}^{2} - {( \sqrt{2}) }^{2} } \\ = \frac{14}{5 - 2} = \frac{14}{3} \\ \\ xy = \frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3} } \times \frac{ \sqrt{5} - \sqrt{3} }{ \sqrt{5} + \sqrt{3} } \\ = 1

Now..

x² + xy + y²

= x² + 2xy + y² - xy

= (x +y) ² - xy

= {( \frac{14}{3}) }^{2} - 1 \\ = \frac{196}{9} - 1 \\ = \frac{196 - 9}{9} = \frac{187}{9}

Hope this is ur required answer

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