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Math Help - number of pair of integers (x,y) satisying an equation

  1. #1
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    number of pair of integers (x,y) satisying an equation

    The equation is
    <br />
x^2-4xy+5y^2+2y-4=0

    What is the total number of pair of integers (x,y) that satisfy the above equation?

    How do I solve it?
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  2. #2
    MHF Contributor red_dog's Avatar
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    (x^2-4xy+4y^2)+(y^2+2y+1)-5=0\Rightarrow(x-2y)^2+(y+1)^2=5

    But 5=1+4=4+1

    Can you continue?
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  3. #3
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    Do I have to test pairs like (2,0), (0,2), (6,2), etc. or there is some other methods?
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  4. #4
    MHF Contributor red_dog's Avatar
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    \left\{\begin{array}{ll}(x-2y)^2=1\\(y+1)^2=4\end{array}\right.

    From here you have to solve four systems:

    \left\{\begin{array}{ll}x-2y=1\\y+1=2\end{array}\right., \ \left\{\begin{array}{ll}x-2y=-1\\y+1=2\end{array}\right., \ \left\{\begin{array}{ll}x-2y=1\\y+1=-2\end{array}\right., \ \left\{\begin{array}{ll}x-2y=-1\\y+1=-2\end{array}\right.

    and to keep the integer solutions.

    Similarly for \left\{\begin{array}{ll}(x-2y)^2=4\\(y+1)^2=1\end{array}\right.
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