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Thread: Find the ordered pairs

  1. #1
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    Find the ordered pairs

    Hello, I'm having some trouble with the following question, and have not been able to solve it despite excessive attempts:

    Find all ordered pairs of integers (x,y) such that  x^2+2x+18=y^2

    I tried square-rooting both sides so that the equation could be solved for y, but didn't have any luck. I also thought about using the "completing the square" method, but am not sure on how to use it in this case. I would greatly appreciate any help.

    Thanks!
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  2. #2
    MHF Contributor

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    I found one pair. There may be others.
    Look at (x+1)^2=y^2-17.
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  3. #3
    MHF Contributor

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    Plato's suggestion, that you rewrite the equation as x^2+ 2x+ 1- 1+ 18= y^2 so that (x+ 1)^2= y^2- 17 is excellent! If you let z= x+1, then that equation becomes z^2= y^2- 17 or y^2- z^2= (y- z)(y+ z)= 17. Since y and z are integers, you should be able to quickly find Plato's solution as well as the three other ordered pairs satifying that equation.
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