The paper presents two necessary and sufficient existence conditions of a common quadratic Lyapunov function for a set of second-order continuous-time linear time-invariant systems. The first condition reduces Lyapunov matrix inequalities to simpler algebraic inequalities containing two quantified variables. Based on the convex property of solution sets of a common diagonal matrix of second-order Lyapunov inequalities and Helly's theorem, the existence condition for a pair of second-order systems is extended to a set of second-order systems. The problem is reduced to algebraic inequalities with a single variable. The obtained results make the problem amenable to QE (Quantifier Elimination) approach. Several examples are given to illustrate the applications of the obtained results.