If a + b = 8 and a^2 + b^2 = 34, what is the value of the product ab?

Introduction / Context:
This algebra question checks your understanding of the identity that relates the square of a sum to the sum of squares and the product of two numbers. Specifically, it uses the identity (a + b)^2 = a^2 + 2ab + b^2. Given a + b and a^2 + b^2, you can use this identity to find the product ab without solving explicitly for a and b.

Given Data / Assumptions:

• a + b = 8.
• a^2 + b^2 = 34.
• We must find ab.

Concept / Approach:
The key identity is (a + b)^2 = a^2 + 2ab + b^2. Since both a + b and a^2 + b^2 are known, you can substitute these values into the identity and solve for ab. This avoids the need to find the individual values of a and b and gives ab directly in a single step.

Step-by-Step Solution:

Verification / Alternative check:
You can check by finding actual numbers a and b that satisfy these conditions. Suppose a and b are the roots of the quadratic t^2 - 8t + 15 = 0, where sum of roots is 8 and product is 15. The roots are 3 and 5. Then a^2 + b^2 = 3^2 + 5^2 = 9 + 25 = 34 and a + b = 3 + 5 = 8. This confirms that ab = 15 is consistent with the given data.

Why Other Options Are Wrong:

Common Pitfalls:
Many students mistakenly apply the identity as a^2 + b^2 = (a + b)^2 + 2ab, which has the wrong sign. The correct relationship is a^2 + b^2 = (a + b)^2 - 2ab. Using the derived equation 64 = 34 + 2ab ensures that the signs are handled properly and leads to the correct product ab = 15.

Final Answer:
15