Question

if X equals to root 5 minus 2 by root 5 + 2 and y equals to root 5 + 2 by root 5 minus 2. then find the value of x square + y square minus xy​

Answer 1

Answer:

Step-by-step explanation:

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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}  

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Answer 2

hi pls give the ans I need it