We present a symbolic approach towards solving the Bergman three-state minimal patient model of glucose metabolism. Our work first translates the Bergman three state minimal patient model into the modified control algebraic Riccati equation. Next, the modified control algebraic Ricatti equation is reduced to a system of polynomial equations, and an optimal (minimal) solution of these polynomials is computed using Dixon resultants. We demonstrate the use of our method by reporting on three case studies over glucose metabolism.