The rate of change of a certain population is proportional to the square root of its size. Model this situation with a differential equation.

The rate of change of a certain population is proportional to the square root of its size. Model this situation with a differential equation.

The solution is said to be $\dfrac{dP}{dt} = k\sqrt{P}$, where $k > 0$ is the proportionality constant.

Since the problem statement says rate of change instead of rate of increase, isn't it true that $k$ can be both positive and negative rather than just positive?

It certainly could be negative. The person who wrote the solution probably mean $|k| > 0$ instead, because if $ k = 0 $, then there is no growth at all. But a negative k is perfectly valid.