Quantum key distribution uses public discussion protocols to establish shared secret keys. In the exploration of ultimate limits to such protocols, the property of symmetric extendibility of underlying bipartite states plays an important role. A bipartite state is symmetric extendible if there exits a tripartite state, such that the AB marginal state is identical to the AB′ marginal state. For a symmetric extendible state, the first task of the public discussion protocol is to break this symmetric extendibility. Therefore to characterize all bipartite quantum states that possess symmetric extensions is of vital importance. We prove a simple analytical formula for a two-qubit state to admit a symmetric extension. Given the intimate relationship between the symmetric extension problem and the quantum marginal problem, our result also provides the first analytical necessary and sufficient condition for the quantum marginal problem with overlapping marginals.