An algebra problem by Julian Yu

Algebra Level 4

Let \(a\), \(b\), and \(c\) be real numbers such that \(a+b+c=2\) and \({a}^{2}+{b}^{2}+{c}^{2}=12\).

If the difference between the maximum and minimum values of \(c\) can be expressed as \(\dfrac{p}{q}\), where \(p\) and \(q\) are coprime positive integers, find \(p+q\).

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