In the current study, we introduce two methods to extract model-physical parameters of a solar panel using photovoltaic metrics at key points. We use a single-diode circuit to describe a solar panel working under STC. According to the first method, we derive a new transcendental equation connecting series resistance R-s to quality factor eta and photovoltaic metrics at key points. Following the second method, we establish a new analytic expression giving series resistance as a function of quality factor and key points coordinates. For both methods, we express parallel conductance G(p), photo-generation current I-ph and leakage current I-s in terms of quality factor and all photovoltaic metrics. We use quality factor as variational parameter to minimize RMSE between experimental and optimized characteristics. We borrow temperature coefficients from manufacturer data sheet and determine panel radiation coefficients to derive analytical expressions giving variations of photovoltaic metrics at key points as functions of panel junction temperature T and solar radiation S. To investigate dependencies of all model-physical parameters versus T and S, we consider numerical values of model-physical parameters at STC as initial conditions, and resolve the system of non-linear equations linking panel current to panel voltage at key points. We test numerical models established and mathematical expressions derived by specifying to PV solar panels such as KC130GT and SM55 operating at different environmental conditions of ambient temperature and solar radiation. As a result, for arbitrary environmental conditions, predicted characteristics agree with measured characteristics validating thereby our numerical approach.