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In this paper, a complete analytical description for an exact expression for temperature dependence of the majority carrier in a single-impurity, nondegenerately doped equilibrium semiconductor is proposed. Analysis establishes that the problem is solvable exactly by identifying the only physically possible root to a cubic equation. This solution is complemented by an iterative technique that identifies boundaries for the intrinsic, freeze-out, and exhaustion regimes and facilitates selecting a reasonable range of temperatures in which to display the exact solution. Similarly, an exact expression for the temperature-dependent Fermi level is obtained. Fairly simple tests and checks on the analytic results are explained and demonstrated. This model provides an attractive alternative or supplement to established classroom approaches for this topic usually covered in senior and first-year graduate-level solid-state courses in physics and electrical engineering.