# Maximizing in a Real Quadratic

Algebra Level 5

$$a, b$$ and $$c$$ are real values satisfying $$a + 2b + 3c = 195$$ and $$a^2 + b^2 + c^2 = 2925$$. The maximum value of $$a$$ has the form $$\frac {p}{q}$$, where $$p$$ and $$q$$ are positive, coprime integers. What is the value of $$p + q$$?

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