The cases of constant and quadratic damping of free oscillations are missing from standard textbooks, even at college and university level. The case most examined is that of linear damping, the reason being that the student can work out a closed form which describes all stages of motion. The case of constant damping is straightforward to be solved, yet it has the inherent difficulty of being described by a different differential equation for each stage of motion. The case of quadratic damping is described by a non-linear differential equation and is not solvable analytically. Despite the above, both cases correspond to real-life conditions. This article presents simple methods for deriving solutions to these problems with tools which any pre-university or first-year undergraduate student can use and work out. Constant damping is solved analytically, while quadratic damping is solved numerically.