CMIMC 2017 Team Problem 10

The polynomial P(x)=x^{3}-6 x-2 has three real roots, \alpha, \beta, and \gamma. Depending on the assignment of the roots, there exist two different quadratics Q such that the graph of y=Q(x) pass through the points (\alpha, \beta), (\beta, \gamma), and (\gamma, \alpha). What is the larger of the two values of Q(1)?