# Thread: Find the ordered pairs

1. ## 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 $\displaystyle x^2+2x+18=y^2$

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!

2. I found one pair. There may be others.
Look at $\displaystyle (x+1)^2=y^2-17$.

3. Plato's suggestion, that you rewrite the equation as $\displaystyle x^2+ 2x+ 1- 1+ 18= y^2$ so that $\displaystyle (x+ 1)^2= y^2- 17$ is excellent! If you let z= x+1, then that equation becomes $\displaystyle z^2= y^2- 17$ or $\displaystyle y^2- z^2= (y- z)(y+ z)= 17$. 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.