Consider a quadratic polynomial \( P(x) = x^2 - sx + p \) where \( r_1 \) and \( r_2 \) are the roots. If \( r_1 + r_2 = 6 \) and \( r_1r_2 = 9 \), use Vieta's formulas and symmetric functions of roots to find the value of \( r_1^2 + r_2^2 \). Let's think step by step and output the final answer within \boxed{}.
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