Treść zadania

Najlepsze rozwiązanie

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  ewka13 17.10.2011 (18:06)

  a)
  (x - y)(x + y) +(x + 2y)(x - 2y)=x^{2} - y^{2} + x^{2} - 4y^{2}=2x^{2} - 5y^{2}
  
  dla :
  x = \sqrt {2} , y = 1 - \sqrt {3}
  
  2(\sqrt {2})^{2} - 5(1 - \sqrt {3})^{2}=4 -5(1 - 2\sqrt {3} + 3)=4 -5(4 - 2\sqrt {3})=
  
  =4 - 20 + 10\sqrt {3}=-16 + 10\sqrt {3}=2(5\sqrt {3} - 8)
  
  b)
  (3x + y)(3x - y) - (x -5y)(x + 5y) =9x^{2} - y^{2} -(x^{2} -25y^{2})=
  
  =9x^{2} - y^{2} - x^{2} + 25y^{2}=8x^{2} + 24y^{2}
  
  dla :
  x = 1+ \sqrt {3} , y = 1+\sqrt {2}
  
  8(1+\sqrt {3})^{2} + 24(1+ \sqrt {2})^{2}=8(1 + 2\sqrt {3} + 3) + 24(1 + 2\sqrt {2} +2)=
  
  =8(4 + 2\sqrt {3}) + 24(3+ 2\sqrt {2})=8[4 + 2\sqrt {3} +3(3+2\sqrt {2})]=
  
  =8(4 + 2\sqrt {3} + 9 + 6\sqrt {2})= 8(13 + 6\sqrt {2} + 2\sqrt {3}).

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