We consider the distribution of roots to a general third-order exponential polynomial equation and give detailed conditions about when all roots lie on the left half plane,a pair of roots cross the imaginary axis and enter the right half plane.These results can be used to discuss the local stability and Hopf bifurcation of three-dimensional biological systems with delay.We apply our results to the three-species delayed food chain models,delayed models for the control of testosterone secretion,delayed models of within-host HIV infection of CD4+T-cells,glucose-insulin systems with delay,and tumor-immune system interaction models with delay.