Explicit approximations for determining the roots of the transcendental equations that arise in the series solutions for one-dimensional transient conduction with one convection and one adiabatic boundary condition are presented for rectangular, spherical, and cylindrical coordinate systems. The method is described for the case of rectangular coordinates so that it can be applied to other coordinate systems, or other combinations of boundary conditions. The error in the root from using the explicit approximations is examined, along with the resulting error in the calculated dimensionless temperature.