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Question
If x = `(7 + 4sqrt(3))`, find the value of
`x^2 + (1)/x^2`
Solution
`x^2 + (1)/x^2`
`(x^2 + (1)/x^2) = (x + (1)/x)^2 - 2` ----(1)
We first find out `x + (1)/x`
`x + (1)/x = (7 + 4sqrt(3)) + (1)/((7 + 4sqrt(3))`
= `((7 + 4sqrt(3))^2 + 1)/((7 + 4sqrt(3))`
= `(49 + 48 + 56sqrt(3) + 1)/((7 + 4sqrt(3))`
= `(98 + 56sqrt(3))/((7 + 4sqrt(3))`
= `(14(7 + 4sqrt(3)))/((7 + 4sqrt(3))`
= 14
substitutingin (1)
`(x^2 + (1)/x)^2 = (x + (1)/x)^2 -2`
= 196 - 2
= 194
∴ `(x^2 + (1)/x^2)` = 194
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