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Question
If x = `sqrt(5) + 2`, then find the value of `x^2 + 1/x^2`
Solution
`sqrt(5) + 2` ⇒ x2 = `(sqrt(5) + 2)^2`
= `(sqrt(5))^2 + 2 xx 2 xx sqrt(5) + 2^2`
= `5 + 4sqrt(5) + 4`
= `9 + 4sqrt(5)`
`1/x = 1/(sqrt(5) + 2)`
= `(sqrt(5) - 2)/((sqrt(5) + 2)(sqrt(5) - 2))`
= `(sqrt(5) - 2)/((sqrt(5))^2 - 2^2)`
= `(sqrt(5) - 2)/(5 - 4)`
= `sqrt(5) - 2`
`1/x^2 = (sqrt(5) - 2)^2`
= `(sqrt(5))^2 - 2 xx sqrt(5) xx 2 + 2^2`
= `5 - 4sqrt(5) + 4`
= `9 - 4sqrt(5)`
∴ `x^2 + 1/x^2 = 9 + 4sqrt(5) + 9 - 4sqrt(5)` = 18
The value of `x^2 + 1/x^2` = 18
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