In this paper, the five parameters of the solar cell single-diode model are analytical and quasi-explicitly extracted for the first time, just using the coordinates of four arbitrary points of the characteristic I-V, curve and the slopes of the curve in these points. The new method presented, called Analytical and Quasi-Explicit (AQE) method, is exact because no simplifications of the model nor a priori approximations of the parameters are used and, it is quasi-explicit in the sense that all the parameters except one are explicitly given. The unique parameter not explicitly computed is easily obtained by solving a five degree polynomial equation. Accurate and practical conditions are provided to select which solution of the previous equation is the desired parameter. It is also introduced a very easy method to obtain, directly-from real data measurements, the needed four points of the I-V curve as well as the slopes in these points, without using any kind of sophisticated techniques. Finally, some experimental results are presented to demonstrate the high accuracy and simplicity of the new method. The results are compared with the well-known analytical five-point method and the recent oblique asymptote method. (C) 2016 Elsevier Ltd. All rights reserved.