From Complex to Real Numbers: A Reverse Detour for Solving Polynomial Equations Using Complex Numbers

Solving polynomial equations by starting with complex numbers appears counter-intuitive particularly when real roots of equations are sought after. However, when attempting to solve polynomial equations such as cubic equations, complex numbers finally appear in the solution even if the roots are all real numbers. Cardan’s solution as such proceeds from real to complex numbers. This paper demonstrates that by starting with complex numbers, it is possible to arrive at the solution that eventually appears in real number form. In effect, such a procedure follows a reverse detour, i.e., from complex to real numbers. In addition, certain factors are simple when expressed in complex forms. This paper presents methods of solving quadratic, cubic, and quartic equations using complex numbers. The formulation of the method and application of the formulae based on the roots of complex numbers is simple and intuitive to follow. Examples are provided for the application of the methods for solving polynomial equations of degrees less than five. The method shows the power of using complex number arithmetic in solving equations despite the fact that the solution can be a real number.