Let \(f: [2, \infty) \mapsto [8, \infty)\) be a surjective function given by

\[\large f(x) = x^2 - (p-2) x + (3p - 2) \]

where \(p \in \mathbb{R}\). If the sum of all possible values of \(p\) can be represented as \(a + b \sqrt{c}\) where \(a, b\) and \(c\) are positive integers and \(c\) is square free, then evaluate \(\dfrac{a}{bc}\).

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