Inductor- and transformer-coupled balanced mixers are usually analyzed with switching approximations, conversion matrices, harmonic balance, or circuit simulation. This paper instead derives their loaded quasistatic equations from the device laws, keeping the transformer ratios, reflected load, bias, and the connections that set the IF common mode. The functions needed to solve these equations are surprisingly varied. A Shockley single-balanced pair is solved by the principal Lambert function W₀ under a quantified one-diode-dominance condition. The double-balanced ring has a unique loaded solution that reduces to an exact inverse-hyperbolic-tangent form only for an open IF port. Vacuum diodes and triodes lead to Cardano cubics and structured sextics, whereas depletion JFET/MESFET crossbars need only square roots. Bilateral MOS channels give piecewise quadratics in strong inversion, or a Ferrari quartic when the IF winding floats. In weak inversion they give an exact W₀, the real r-Lambert function, or the inverse of sinh x+ρ x, depending on topology and IF connection. In matched BJT and HBT Gilbert cells, equal emitter resistors turn the hyperbolic-tangent current steering into an exact r-Lambert function. Fourier projection brings in Beta/Gamma, modified Bessel, and complete elliptic functions. We checked the equations against 37 LTspice simulations using seven independent Python programs. In the 12 primary comparisons, which implement the assumed device laws exactly, every RMS waveform error was below 0.017% of the analytical RMS value; built-in and vendor device models were examined separately. Two design examples show how the closed forms size a thermionic IF load and a Gilbert-cell tail current without parameter sweeps.