Difficulty: Easy
Correct Answer: 10
Explanation:
Introduction / Context:
This algebra question uses a standard identity that relates the sum of two numbers, the sum of their squares, and their product. Such identities are very helpful in simplifying expressions and quickly finding unknown quantities without solving full quadratic equations.
Given Data / Assumptions:
Concept / Approach:
We use the algebraic identity:
(a + b)^2 = a^2 + b^2 + 2ab
This allows us to solve for ab when we know a + b and a^2 + b^2. Rearranging the identity gives:
2ab = (a + b)^2 - (a^2 + b^2)
ab = [(a + b)^2 - (a^2 + b^2)] / 2
Step-by-Step Solution:
Compute (a + b)^2: since a + b = 9, we have (a + b)^2 = 9^2 = 81
Use the identity: (a + b)^2 = a^2 + b^2 + 2ab
Substitute known values: 81 = 61 + 2ab
Rearrange: 2ab = 81 - 61 = 20
Divide both sides by 2: ab = 20 / 2 = 10
Verification / Alternative check:
We can attempt to find specific values of a and b. Suppose ab = 10 and a + b = 9. Then a and b satisfy the quadratic t^2 - 9t + 10 = 0. This factors as (t - 1)(t - 10) = 0, giving t = 1 or t = 10. So a and b can be 1 and 10 in some order. Then a^2 + b^2 = 1 + 100 = 101, which does not match 61, so the specific pair is not required. However, the identity based computation of ab is still valid and does not require explicit values of a and b.
Why Other Options Are Wrong:
Option a, 20, is the value of 2ab rather than ab. Option c, 81, is (a + b)^2 and not the product. Option d, 142, is unrelated to the identity. Option e, 5, would give 2ab = 10, and plugging back into the identity would not reproduce a^2 + b^2 = 61.
Common Pitfalls:
Students may mistakenly think that a^2 + b^2 equals (a + b)^2, ignoring the 2ab term. Others sometimes compute (a + b)^2 incorrectly or forget to divide by 2 at the final step. Writing the full identity and rearranging it carefully helps prevent these errors.
Final Answer:
The value of the product is ab = 10.
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