Question

Consider the quadratic function \( f(x) = ax^2 + bx + a \) having two irrational roots, with \( a \) and \( b \) being two positive integers, such that \( a, b \leq 9 \). If all such permissible pairs \( (a, b) \) are equally likely, what is the probability that \( a + b \) is greater than 9?

• A. \( \frac{5}{8} \)
• B. \( \frac{2}{3} \)
• C. \( \frac{7}{15} \)
• D. \( \frac{7}{16} \)

Hint:

Use the discriminant to identify conditions for irrational roots, and carefully count the pairs that satisfy the given condition.