If a and b are real numbers such that a + b = 11 and a^2 + b^2 = 61, then using standard algebraic identities, what is the value of the product ab?

Introduction / Context: This question checks your understanding of the identity relating the square of a sum to the sum of squares and the product of two numbers. Instead of solving for a and b individually, you can use the relation (a + b)^2 = a^2 + 2ab + b^2 and the given values of a + b and a^2 + b^2 to find ab quickly. This is a common pattern in algebra questions on aptitude exams.

Given Data / Assumptions:

• a + b = 11.
• a^2 + b^2 = 61.
• a and b are real numbers.
• We are asked to find ab.

Concept / Approach: Use the identity (a + b)^2 = a^2 + 2ab + b^2. We already know a + b and a^2 + b^2, so substituting these into the identity gives an equation in terms of ab. From this, we can isolate 2ab, find its value, and then divide by 2 to get ab. There is no need to find explicit values of a and b, though that is also possible in principle.

Step-by-Step Solution: Start from the identity (a + b)^2 = a^2 + 2ab + b^2. We are given a + b = 11, so (a + b)^2 = 11^2 = 121. We are also given a^2 + b^2 = 61. Substitute into the identity: 121 = 61 + 2ab. Rearrange to solve for 2ab: 2ab = 121 - 61. Compute the difference: 121 - 61 = 60. So 2ab = 60. Divide both sides by 2: ab = 60 / 2 = 30.

Verification / Alternative check: We can verify by constructing a and b as roots of a quadratic equation. If a and b are roots of t^2 - (a + b)t + ab = 0, then the quadratic is t^2 - 11t + 30 = 0. Factor this: t^2 - 11t + 30 = (t - 5)(t - 6). So a and b could be 5 and 6. Then a + b = 11 and a^2 + b^2 = 25 + 36 = 61, both matching the given values. The product ab = 5 * 6 = 30, confirming our result.

Why Other Options Are Wrong: Values like 12, 24, 18, or 96 would not satisfy the relationship 2ab = 60. If any of those were used as ab, then a^2 + b^2 computed from (a + b)^2 - 2ab would not equal 61. For example, if ab were 24, then a^2 + b^2 would be 121 - 48 = 73, which contradicts the given 61.

Common Pitfalls: A common mistake is to misapply the identity as a^2 + b^2 = (a + b)^2 + 2ab instead of (a + b)^2 - 2ab. Another error is in arithmetic when squaring 11 or subtracting 61 from 121. Carefully writing the identity and substituting the given values avoids these issues.

Final Answer: The value of the product ab is 30.