A Prime Quadratic

Let \(S\) be the set of all ordered triples \((p,q,r)\) of prime numbers for which at least one rational number \(x\) satisfies \(px^2+qx+r=0\). Find all the primes that appear in seven or more elements of \(S\)?

Enter your answer as the sum of all such primes.
Enter 0 if there is no such prime.

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