Oscillations of a system with strong quadratic damping are considered. For the exact analytical form of the energy-displacement function the explicit form of the maximal amplitudes of vibration are obtained by introducing the Lambert W function. Comparing the neighbor maximal amplitudes and the corresponding energies the conclusions about the energy dissipation is given. The approximate solution for a strong nonlinear differential equation which describes the motion of the oscillator with quadratic damping is calculated applying the elliptic-harmonic-balance method. The accuracy of the solution is affirmed by comparing the maximal displacements obtained using the approximate method with the exact one obtained by energy method.