A Heisenberg ferromagnet with biquadratic exchange and single-ion anisotropy is studied by the irreducible Green function formalism. Equations of motion for different Green functions are solved by means of irreducible operators and the exact matrix Dyson equation is derived. The low-temperature results for spin-wave damping and energy shift are discussed. It is found that the spin-wave modes in the quadrupolar state of the system disappear under certain conditions. The expressions for such modes are also found. The variation of Curie temperature with respect to the strength of the biquadratic exchange and single-ion anisotropy is also studied.