### Instruction: For how many different negative values of $x$ is $\sqrt{x +150}$ a positive integer?### Response: So we're looking for negative values of $x$ that would make $\sqrt{x+150}$ a positive integer?
Right.
So that would mean that $x+150$ would have to be a perfect square.
Right.
So we could write this as $x+150=n^2$, where $n$ is a positive integer.
And we're looking for negative values of $x$, so $x=n^2-150$.
So $n^2-150<0$.
So $n^2<150$.
So $n<12$.
So $n$ can be $1, 2, 3, ..., 11$.