If \(x^2+ax+b \epsilon Z\) for any \(x \epsilon Z\) and x2+ax+b=0 has rational roots then prove that the same roots are integers.
For all integral values of x it's given that x2+ax+b is integer. Putting x=0 it can be seen that b is an integer. ax+b must be an integer for all integral values of x and since b is integer, a must be integer as well.
a2-4b is perfect square but since a and b are integers it must be square of integer. Clearly, both roots must be integers.