If a - b = 3 and a^2 + b^2 = 29 for real numbers a and b, what is the value of the product ab?

Introduction:
This question tests using algebraic identities to connect a - b, a^2 + b^2, and ab. The key identity is (a - b)^2 = a^2 + b^2 - 2ab. Since (a - b) and a^2 + b^2 are both given, we can compute (a - b)^2 and then solve for ab. This avoids solving for a and b individually, which would be longer and unnecessary. It is a common simplification technique in algebra and aptitude exams.

Given Data / Assumptions:

• a - b = 3
• a^2 + b^2 = 29
• We need to find ab
• Identity used: (a - b)^2 = a^2 + b^2 - 2ab

Concept / Approach:
Square the difference to get (a - b)^2. Then substitute into the identity (a - b)^2 = a^2 + b^2 - 2ab. With a^2 + b^2 known, the only unknown becomes ab. Solve the resulting linear equation for ab.

Step-by-Step Solution:
Given a - b = 3, so (a - b)^2 = 3^2 = 9 Use identity: (a - b)^2 = a^2 + b^2 - 2ab Substitute values: 9 = 29 - 2ab 2ab = 29 - 9 = 20 ab = 20/2 = 10

Verification / Alternative check:
We can also verify consistency by finding possible a and b. From a - b = 3, let a = b + 3. Then a^2 + b^2 = (b+3)^2 + b^2 = 2b^2 + 6b + 9 = 29. So 2b^2 + 6b - 20 = 0 => b^2 + 3b - 10 = 0 => (b+5)(b-2)=0, so b=2 or b=-5. Corresponding a=5 or a=-2. In both cases, ab = 10. Verified.

Why Other Options Are Wrong:
Any value other than 10 would violate the identity relation because (a - b)^2 and a^2 + b^2 are fixed. The equation 9 = 29 - 2ab forces ab to be 10 exactly.

Common Pitfalls:
Using (a + b)^2 instead of (a - b)^2, forgetting the -2ab term, or mis-subtracting 29 - 9.

Final Answer:
The value of ab is 10.