Quadratic B$acute{e}$zier curves are important geometric entities in many applications. However, it was often ignored by the literature the fact that a single segment of a quadratic B$acute{e}$zier curve may fail to fit arbitrary endpoint unit tangent vectors. The purpose of this paper is to provide a solution to this problem, i.e., constructing $G^1$ quadratic B$acute{e}$zier curves satisfying given endpoint (positions and arbitrary unit tangent vectors) conditions. Examples are given to illustrate the new solution and to perform comparison between the $G^1$ quadratic B$acute{e}$zier cures and other curve schemes such as the composite geometric Hermite curves and the biarcs.