The aim of this study is to examine the argumentation-based problem solving processes of 10th grade students in the context of quadratic equations. The study was carried out in a public high school in Istanbul. The study group consists of three male students selected by the method of "homogeneous sampling". In the research, special case study, one of the qualitative research models, was used. The data in the application process were collected with two problem-solving activities related to quadratic equations. The students tried to solve the problems together by doing group work with one week interval for both problems. The problem solving processes of the students were recorded with the video recorder and the discussions between the students were analyzed according to Toulmin's argumentation model. According to the findings of the research, it was determined that the students used the argument (65), least rebuttal (3) and qualitative (3) components among the argumentation components. Students produced less arguments in the first problem solutions than the second problem. For this reason, differences were observed in students' ability to produce arguments according to the qualifications of the questions. In addition, after the argumentation study, the students stated that the problem solving process contributed in the context of understanding the problem for themselves, expressing their thoughts, persuading their opinions by defending their ideas, and communicating together and they want to be found again in similar working environments.