The neutron transport equation for one-speed neutrons in a homogeneous slab was solved for the solution of the extrapolation distance problem. In the solution algorithm of the problem first the traditional P1 method in which the neutron angular flux was expanded in terms of the Legendre polynomials and then the U1 method in which the neutron angular flux was expanded in terms of the Chebyshev polynomials of second kind were used. While it is possible to find the solution of the problem obtained by P1 method in literature, U1 method was first applied to this problem and an analytical expression was obtained as the main goal of this study. Finally, numerical results for the extrapolation distances were calculated using both methods and they were given in the tables together with the exact results for comparison. Although the results obtained in this study could be evaluated as not to be in a very good accordance with the results already presented in the literature; this should be realised as these calculations were carried out using only the lowest order approximations of the polynomial expansion techniques. Then it is clear that more convenient results with the literature may be obtained in case applying higher-order aproximations.However, it is important to underline that the U1 method was first pplied to the problem and an analytical expression for the extrapolation distance was derived explicitly.