How are these equations for energy compatible? I have one equation that says the energy in an oscillating spring with a mass
$$E = 2\pi^2ν^2A^2M.$$
I have another equation that says the energy is the system is quantized
$$E = nhν.$$
How are these equations compatible? Why is there a factor of $ν^2$ in one but $ν $ in another?
 A: Energy is measured in Joules. 
For your second equation the units are:
$$\left[h\right]=J\cdot s$$
$$\left[\nu\right]=s^{-1}$$
$n$ is a number.
For the first equation, if $M$ is the mass of the spring, then (based on $E=mc^2$):
$$\left[\nu^2\cdot A^2\right]=s^{-2}\cdot m^2$$
$2\pi^2$ is just a constant.
So it is possible as long as the dimensions are correct.
A: First, $\nu$ is a constant, a property of the oscillator, just like its mass. Two equations for the oscillator's energy need not agree in their dependency on it because it does not vary.
Second, the first equation relates energy to the square of the amplitude. The second equation to the first power of some abstract number $n$. All other symbols ($M$, $h$ etc.) are constants. So there's no contradiction here either. It just means that there is a proportion between $A^2$ and $n$, giving an idea of interpretation of this $n$ (or, more precisely, how $A$ relates to $n$).
What the two equations mean is that the energy, although (classically) expressed as a continuous function of amplitudes, can in fact (quantum-ly) only take on discrete values, multiples of $h\nu$. That is possible to reconcile if we postulate that (semiclassically) only some amplitudes of the oscillator are permitted by some hidden mechanism. This is basically, in a very simplified manner, the result of Bohr and Sommerfeld.
You can find (again, semiclasically: this is not a result in quantum mechanics and not classical physics either) the allowed amplitudes simply by putting an "=" between the two formulas:
$$\begin{aligned}
2\pi^2\nu^2A^2M &= nh\nu \\
A^2 &= \frac{nh}{2\pi^2\nu M} \\
A &= \sqrt\frac{nh}{2\pi^2\nu M} \\
\end{aligned}$$
One can see that $A$, as a function of $n\in\mathbb{N}$, depends on $\nu$ just as it depends on the mass of the oscillator. Lighter or slower oscillators would tend to swing farther from the origin even in low quantum numbers. In quantum mechanics, the corresponding finding is that they have a higher statistical variation in position at the same $n$.
BTW. The correct quantization of energy is slightly different,
$$E = h\nu\left(n+\frac12\right),$$
and allows $n=0$ as well. The difference between two adjacent energy levels remains $h\nu$, just the zero point energy is different.
