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.

Answer 1

The Correct Option is B

Step 1: Condition for irrational roots.
For a quadratic to have irrational roots, the discriminant must be positive and not a perfect square. The discriminant for \( f(x) = ax^2 + bx + a \) is: \[ \Delta = b^2 - 4ac = b^2 - 4a^2 \] We require \( \Delta \) to be positive and not a perfect square.
Step 2: Count permissible values of \( a \) and \( b \).
There are \( 9 \) possible values for \( a \) and \( b \) (since both \( a \) and \( b \) are positive integers less than or equal to 9).
Step 3: Calculate probability.
The probability is based on the number of permissible pairs \( (a, b) \) such that \( a + b>9 \).
Final Answer: \[ \boxed{\frac{2}{3}} \]