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Question
Show that: `x^2 + 1/x^2 = 34,` if x = 3 + `2sqrt2`
Solution
Given: x = 3 + `2sqrt2`
`1/x = 1/(3 + 2sqrt2) xx (3 - 2sqrt2)/(3 - 2sqrt2)`
`1/x = (3 - 2sqrt2)/((3)^2 - (2sqrt2)^2)`
`1/x = (3 - 2sqrt2)/(9 - 8)`
`1/x = 3 - 2sqrt2`
Now, `x + 1/x = 3 + cancel(2sqrt2) + 3 - cancel(2sqrt2)`
`x + 1/x = 6`
Squaring on both sides
`(x + 1/x)^2 = (6)^2`
`=> x^2 + 1/x^2 + 2 xx cancel(x) xx 1/cancel(x) = 36`
`=x^2 + 1/x^2 = 36 - 2`
`= x^2 + 1/x^2 = 34`
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