If x = 1 / (√2 - 1), then x - 1/x = ____?
- 2
- √2
- 2√2
- None of these
Explanation
Given:
x = 1 / (√2 - 1)
To simplify, multiply the numerator and denominator by (√2 + 1):
x = 1 / (√2 - 1) × (√2 + 1) / (√2 + 1)
= (√2 + 1) / (√2² - 1²)
= (√2 + 1) / (2 - 1)
= (√2 + 1)
Now, find x - 1/x:
x - 1/x = (√2 + 1) - 1 / (√2 + 1)
= (√2 + 1) - (√2 - 1) / (√2 + 1)(√2 - 1)
= (√2 + 1) - (√2 - 1) / (2 - 1)
= (√2 + 1) - (√2 - 1)
= (√2 + 1 - √2 + 1)
= 2
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