Based on Hamilton’s method of solving couple equation, defined over the field of couple numbers, as he calls them, in this chapter, the method of solving the general Hamilton quadratic equation (couple equation) is reduced to solving of the matrix quadratic equation, which is defined over the field of bireal matrices which correspond to bireal numbers. As the matrix equation is decomposed into a system of nonlinear algebraic equations (SNAE), the solutions of the matrix equation are also the solutions of the SNAE, such that the solutions of the matrix quadratic equation implicitly determine the symmetric matrices S of the null space of the SNAE. In the second part of this chapter, the matrix method for solving the SNAE, which is obtained via decomposition of Hamilton’s quadratic equation, is presented.