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nuox95

X+y = 2 2xy-(z^2)= find the interger solutions of the equation.

  • 2 years ago
  • 2 years ago

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  1. dipankarstudy
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    is it 2xy-z^2=0??

    • 2 years ago
  2. nuox95
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    oh wait , forgot to type it out, no, 2xy-z^2=1

    • 2 years ago
  3. dipankarstudy
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    if it is so ..then 2xy=z^2+1 (x+y)^2=4 x^2+y^2+2xy=4 x^2+y^2+z^2=3 so plot the sphere with radius sqrt(3) and find the points which has integer (x,y,z) triplet

    • 2 years ago
  4. dipankarstudy
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    one such point is (1,1,1)

    • 2 years ago
  5. dipankarstudy
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    only this is the solution..no other points are there.

    • 2 years ago
  6. dipankarstudy
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    u can prove it by this way x+y>=2sqrt(xy) sqrt(xy)<=1 as x+y=2 xy<=1 2xy<=2 z^2+1<=2 z^2<=1 z=1 only integer.. so 2xy=1^2+1=2 xy=1 s0 x=1 y=1 the solution.. (1,1,1) is the only possible solutin

    • 2 years ago
  7. nuox95
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    actually I have the solution to the question which is x+y=2 and 2xy-z^2=1 leads to 2(x-1)^2+z^2=1, hence interger solutions are (1,1,1) and (1,1,-1) but i dont really get it, preparing for a math test which im not par with.

    • 2 years ago
  8. dipankarstudy
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    ok....take my equation x^2+y^2+z^2=3 so it can be x^2=1, y^2=1, z^2=1 now x+y=2>0 so x,y>0 so x=y=1 but z^2=1 gives z=1,-1 so (1,1,1) and (1,1-1)

    • 2 years ago
  9. dipankarstudy
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    got it??

    • 2 years ago
  10. nuox95
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    i get that z can be either -1 ,1 but how did you get x^2+y^2+z^2=3?

    • 2 years ago
  11. dipankarstudy
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    and also from second approach z^2<=1 z^2=1 z=1,-1 so 2xy=2 xy=1 and both x,y>0 so x=y=1

    • 2 years ago
  12. dipankarstudy
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    2xy=z^2+1 (x+y)^2=4 x^2+y^2+2xy=4 x^2+y^2+z^2=3

    • 2 years ago
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