Common Ion Effect Calculations
The value of the answer you were given is incorrect, both in its value and its explanation of what can be ignored.

As you note:

$K_{sp} = 4.90\times10^{-18}=[\ce{Fe^{2+}}][\ce{S^{2-}}]$

Just eyeballing the equation, we would expect a solution of $\ce{FeS}$ in pure water to contain equal quantities of iron(II) and sulfide ions, at a concentration of $2.21\times10^{-9} M$ (the square root of the solubility product.)  This is on the order of a million times more concentrated than the solution described in the problem, so we can ignore the sulfide concentration in the original solution.  For the purposes of this problem, the iron sulfide is dissolving in pure water.  

Additionally, solving the quadratic equation you provided gives much the same result.  The solution rounds to $x = 2.21\times10^{-9} M$; it only differs in the sixth place from the square root of the solubility product.  It's easy enough to make a small numerical error with scientific notation and solving a quadratic; I have to assume that's what happened to you here.

The only way that your instructor's answer would be correct would be if the question intended us to understand that there was some chemistry going on to keep the concentration of sulfide ions at the given concentration.  Without that, it just doesn't make sense.