Let $P_i(x) = x^2 + b_i x + c_i , i = 1,2, ..., n$ be pairwise distinct polynomials of degree $2$ with real coefficients so that for any $0 \le i < j \le n , i, j \in N$, the polynomial $Q_{i,j}(x) = P_i(x) + P_j(x)$ has only one real root. Find the greatest possible value of $n$.