Machine learning involves all the methods used to find a solution to the problems of clustering, classification, regression. In Chapter 1 the basic steps and ideas of machine learning for supervised methods are presented. The goal of the Chapter 2 is to build a theory foundation for the Support Vector Machine (SVM) on the three optimizations theorems (Fermat theorm, Lagrange theorem, Karush-Kuhn-Tucker theorem) that play a major role in finding solution to the SVM. In the Chapter 3, the main concepts on which SVM is based are presented and we remark how to exploit the two great adventages obtained from the dual formulation. The first advantage is that the method in dual formulation is a quadratic programming problem whose solution exists and is known (KKT theorem). The second advantage is the possibility of using kernels, which allow the application of the method to a dataset represented by the features on which is possible to build a separating hyperplane. In the final chapter of this thesis we apply the method which is based on the Sequential Minimal Optimization algorithm. Unlike the other known algorithms for solving problems of SVM, SMO goes to extremes and solves optimization subproblems for only two Lagrange multipliers. This is why SMO has proven to be one of the most effcient algorithms in solving the problem of support vector methods. At the end of the thesis we present the results of the described method applied to the binary classification problem in banking. The goal of this example is to build a method that can be used to identify clients leaving the bank. Once the initial data set has been prepared for the method and recall is selected as a validation measure of the method, we start the training process. Using cross-validation we optimize the hyperparameters of the method on the train data set, and finally we test the generalization power of the method for the optimal parameters on the test data set. The best results are achieved for the method based on the polynomial kernel.