Final answer:
The worst-case runtime for an insertion in a hash table with quadratic probing and given constraints is typically O(1), though it theoretically could be O(n) in an extreme scenario. Such cases are rare with a good hash function and a table that is at most half full.
Step-by-step explanation:
The worst-case runtime for inserting a value in a hash table that uses quadratic probing for collision resolution is O(n). However, under the assumption that the hash table is at most half full, and the capacity is prime and congruent to 3 modulo 4, the worst case is significantly reduced. Since quadratic probing is used and the table's capacity meets these specific conditions, it is designed to reduce the likelihood of primary clustering, increasing the efficiency of the probing. Still, the worst-case scenario of having to probe through many elements could occur if the hash function distributes values poorly or in a worst-case hash sequence. But in practice, such extreme cases are rare, particularly with a well-designed hash function and under the assumption that the array is only half full.
In this case, the worst-case run time would most likely be O(1) due to the aforementioned constraints even though theory still suggests an O(n) possibility in an extreme scenario where every probe location is already filled until the insert succeeds or the table is found to be full.