Quaternion algebra has been widely applied to many subjects such as quantum mechanics, control theory and inertial navigation, and it has won attention from many scholars in the field of effectively obtaining the roots of a quaternionic polynomial. Recently, Janovska and Opfer have theoretically provided a method of finding all zeros of a simple quaternionic polynomial of degree n. Furthermore, Feng and Zhao have given a formula of finding all zeros of a general simple quaternionic polynomial of degree n in terms of solving polynomials over the field of complex numbers. Based on the results given by Feng and Zhao in this paper, the roots of a quaternionic one-sided polynomial with degree 2 were discussed and classified, and a quadratic formula for quaternions with the help of the Ferrari's quadratic formula over the field of complex numbers was produced. Compared with the results in literature, the formula built in this paper displays its advantages in many aspects.