jmc

We could use the quadratic formula, but there is a shortcut: note that if the quadratic is not a perfect square, the solutions will be of the form $p\pm \sqrt{q}$ or $p\pm i\sqrt{q}$. In the first case, if both solutions are real, there are 2 different values of $|z|$, whereas in the second case, there is only one value, since $|p+i\sqrt{q}|=|p-i\sqrt{q}|=\sqrt{p^2+q}$. So all we have to do is check the sign of the discriminant: $b^2-4ac=64-4(37)<0$. Since the discriminant is negative, there are two nonreal solutions, and thus only $\boxed{1}$ possible value for the magnitude.