If a² − b² = 9 and (a − b)² = 3 then ab is equal to
- 3
- 22
- 6
- 18
Explanation
We are given:
1. a² − b² = 9
2. (a − b)² = 3
We can use the identity:
a² − b² = (a − b)(a + b)
From (1):
⇒ (a − b)(a + b) = 9 — (i)
From (2):
(a − b)² = 3
⇒ a − b = √3 or −√3 — (ii)
Substitute (ii) into (i):
If a − b = √3
Then (√3)(a + b) = 9
⇒ a + b = 9 / √3 = 3√3
Now use the identity:
(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b² = 3
Add both:
(a + b)² + (a − b)² = 2a² + 2b² = 2(a² + b²)
⇒ (a + b)² + 3 = 2(a² + b²)
We also know:
a² − b² = 9, and
(a + b)(a − b) = 9
So, use formulas to isolate ab:
From identity:
a² + b² = (a + b)² − 2ab
a² − b² = 9 — already known
Now subtract equations:
(a + b)² − (a − b)² = 4ab
Use known values:
(3√3)² − 3 = 4ab
9×3 − 3 = 4ab
27 − 3 = 4ab
24 = 4ab
⇒ ab = 6
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