Objectives: Boundary value issues occasionally offer a dual solution, one of which is really trustworthy and stable while the other is unstable. The intention of the current research is to use the Casson model to explain the double solutions of graphene/water nanofluid flow through a stretching or compressing wedge and stagnation point. For a specific range of the suction/injection parameter and the extending/contracting parameter, two solutions are found. Methods: The fluid transport equations are then converted into ordinary differential equations by applying the apt self-similarity variables and are computed using MATLAB's bvp4c. Graphs are used to show how certain factors affect the flow field. Findings: The results of the current research shows that the permeability constraint on velocity and thermal variations obtained from the primary and secondary solutions are contrary in nature. The suction/injection parameter and skin friction are directly related for the first solution, whereas a contrary for other. The magnetic parameter reduces the temperature for the initial solution in both cases. According to the stability analysis, the second option is reliable.