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Question
If `x = sqrt(2) + 1`, find `x^2 + 1/x^2`
Sum
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Solution
Given: `x = sqrt(2) + 1`
Stepwise calculation:
1. Find `1/x`:
`1/x = 1/(sqrt(2) + 1)`
= `(sqrt(2) - 1)/((sqrt(2) + 1)(sqrt(2) - 1))`
= `sqrt(2) - 1`
2. Calculate `x + 1/x`:
`x + 1/x = (sqrt(2) + 1) + (sqrt(2) - 1)`
`x + 1/x = 2sqrt(2)`
3. Use the identity:
`(x + 1/x)^2 = x^2 + 2 + 1/x^2`
⇒ `x^2 + 1/x^2 = (x + 1/x)^2 - 2`
Substitute `(x + 1/x = 2sqrt(2)):`
`x^2 + 1/x^2 = (2sqrt(2))^2 - 2`
= 8 – 2
= 6
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Chapter 1: Rational and Irrational Numbers - Exercise 1E [Page 32]
