Two positive real roots

Algebra Level 4

A quadratic equation \( x^{2} -10x -5 = ax + b \) has two positive real roots \( \alpha\) and \(\beta. \) Define \( p \) as the minimum possible integer value of \( a ,\) and define \( q \) as the minimum value of \( \alpha ^{2} + \beta ^{2}. \) What is \( pq? \)

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