If a² - b² = 9 and (a - b)² = 3, then what is the value of ab?
- 2
- 3
- 4
- 6
Explanation
a² − b² = 9
But we can also write:
a² − b² = (a + b)(a − b) = 9
And
(a − b)² = 3 → so (a − b) = √3 or −√3
Let’s assume:
a − b = √3
Then:
From (a + b)(a − b) = 9
⇒ (a + b)(√3) = 9
⇒ a + b = 9 / √3 = 3√3
Now, we have:
a + b = 3√3
a − b = √3
Now solve the system:
Adding:
2a = 4√3 → a = 2√3
Subtracting:
2b = 2√3 → b = √3
Now, ab = a × b = 2√3 × √3 = 2 × 3 = 6
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