Factoring to solve the critically damped harmonic oscillator

When a harmonic oscillator is critically damped, life becomes more complicated because the normal solution based on generalizing the simple harmonic oscillator — that is guessing an exponential solution — only has one root and so one solution. That can’t be correct for a second-order differential equation, so another solution needs to be derived. We did it one way in class, and another method, factoring, is presented here.