If a - b = 3 and a2 + b2 = 29, find the value of ab.

Concept: For any two numbers a and b, the identity (a - b)2 = a2 + b2 - 2ab links the square of their difference, the sum of their squares, and their product. Rearranging it gives 2ab = (a2 + b2) - (a - b)2, so ab can be found directly once a - b and a2 + b2 are both known — no need to find a and b individually.

Applying it here:

1. Square the given difference: a - b = 3, so (a - b)2 = 32 = 9.

2. Use the given sum of squares: a2 + b2 = 29.

3. Substitute both into the identity: 2ab = (a2 + b2) - (a - b)2 = 29 - 9 = 20.

4. Divide by 2 to isolate ab: ab = 20 / 2 = 10.

Cross-check: Since (a + b)2 = a2 + b2 + 2ab = 29 + 20 = 49, a + b = ±7. Taking a + b = 7 with a - b = 3 gives a = 5, b = 2; taking a + b = -7 with a - b = 3 gives a = -2, b = -5. In both cases a × b = 10 — the same value obtained above.