Constructing a \(G^{2}\) continuous curve with the given endpoint conditions is crucial in CAD/CAM applications. Previous literature often transforms this problem into a non-convex optimization problem, typically involving the solution of a system of equations with multiple variables( \(\ge \) 4), where real roots do not always exist. In this paper, we propose an intuitive and efficient method for constructing a quartic Bézier curve with \(G^2\) continuity constraints at the endpoints. Our method consists of two steps: the first step involves selecting the middle control points by geometric method to ensure curve fairness, cusp-free and loop-free. The second step involves determining the optimal curve by applying constraint conditions. Throughout the entire process, we only need to solve a system of quadratic (or linear) equations in two variables. We also provide the conditions under which our system can be solved. Experimental results demonstrate that our algorithm is fast and efficient.