You have been given k is an >positive< integer right??

So use it

Notice that a and b are rationals and roots of x^2 + 3x + k = 0.

And if k>=3 then 9 - 4k < 0 and hence does not exist. This means You have to check only for k=1 and k=2, Notice that which is irrational. But this cant hold because we have been told that roots are rationals. Thus now check for k=1........ And then claim it is the only solution.

All done