if x and y are integers

X and y are integers such that x<y<0. What is x?

1. (x+y) (x-y) = 7
2. xy = 12

For 1. After FOILing the left side and isolating x, we reach
X^2 = y^2 + 7
And then we take the Sq. root of both sides, resulting in
X = Sq. root (y^2+7)

After this GMATfocus simply says that “trial and error will show that the only value of y for which this equation results in an integer is -3, which makes x = -4”.

Trial and error is fine, but I wasn't able to pick the right numbers. Can you think of another method to solve the problem?

Thanks,