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.
A quadratic equation yields irrational roots if its discriminant is positive and not a perfect square. The discriminant for \( f(x) = ax^2 + bx + a \) is calculated as:\[\Delta = b^2 - 4ac = b^2 - 4a^2\]The discriminant \( \Delta \) must satisfy the conditions of being positive and not a perfect square.
Step 2: Determine the count of valid \( a \) and \( b \) values.
Given that \( a \) and \( b \) are positive integers not exceeding 9, there are 9 possible values for each, resulting in 9 potential values for \( a \) and 9 for \( b \).
Step 3: Compute the probability.
The probability is derived from the quantity of valid pairs \( (a, b) \) where the sum \( a + b \) exceeds 9.
Final Answer: \[\boxed{\frac{2}{3}}\]