Any possible rational root must be negative when all the coefficients are positive. The coefficients a and c dictate the possible rational roots have numerator and denominator at most 100. So then we want such a fraction as close to zero as possible on the negative side.
The closest two values are -1/100 and -1/99.
If x=-1/100 then a=100 is mandatory. Then substitute and rearrange a bit to get 1/100 + c = b/100. This would imply b has to be at least 101, which is out of the range. So no solution here.
Then do the same thing with -1/99 and eventually get 1/99 + c = b/99. Here we do get a valid solution with b=100 and c=1.
Solution