jmc

Let $k=\frac{m}{n}$ in reduced form, where $m$ and $n$ are integers. Then by the Rational Root Theorem, $m$ divides 12 and $m$ divides 75, so $m$ must divide $\gcd (12,75)=3.$ Similarly, $n$ divides 75 and $n$ divides 12, so $n$ must divide $\gcd (75,12)=3.$ Thus, $m,$ $n\in \{-3,-1,1,3\}.$

We are told that $k=\frac{m}{n}$ is not an integer, and negative. The only possibility is that $k=\boxed{-\frac{1}{3}}.$