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Question
If \[x = \sqrt{6} + \sqrt{5}\],then \[x^2 + \frac{1}{x^2} - 2 =\]
Options
\[2\sqrt{6}\]
\[2\sqrt{5}\]
24
20
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Solution
Given that `x = sqrt6 +sqrt5 ` .Hence `1/x`is given as
`1/x = 1/(sqrt6+sqrt5)`.We need to find `x^2 +1/x^2 - 2`
We know that rationalization factor for `sqrt6 +sqrt5` is`sqrt6 -sqrt5`. We will multiply numerator and denominator of the given expression `1/(sqrt6+sqrt5)`by `sqrt6 -sqrt5`, to get
`1/x = 1/(sqrt6+sqrt5) xx (sqrt6-sqrt5)/(sqrt6-sqrt5) `
` = (sqrt6-sqrt5)/((sqrt6)^2 - (sqrt5)^2)`
` = (sqrt6 - sqrt5)/(6-5)`
` = sqrt6 - sqrt5.`
We know that `(x-1/x)^2 = x^2 + 1/x^2 - 2 ` therefore,
`x^2 + 1/x^2 - 2 = (x-1/x)^2 `
` = (sqrt 6 + sqrt5 - (sqrt6 - sqrt5))^2`
` = (sqrt6 + sqrt5 - sqrt6 +sqrt5)^2`
` = (2sqrt5)^2`
`= 20`
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