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Question
If \[x + \sqrt{15} = 4,\] then \[x + \frac{1}{x}\] =
Options
2
4
8
1
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Solution
Given that .`x+sqrt15 = 4` It can be simplified as
` x =4 -sqrt15`
`1/x = 1/(4-sqrt15)`
We need to find `x+1/x`
We know that rationalization factor for `4-sqrt15 `is `4 +sqrt15`. We will multiply numerator and denominator of the given expression `1/(4-sqrt15)`by , `4+sqrt15` to get
`1/x = 1/(4-sqrt15) xx (4+sqrt15)/ (4+sqrt15) `
`= (4+sqrt15)/((4^2) - (sqrt15)^2)`
`= (4+sqrt15)/(16-15)`
`= 4+sqrt15`
Therefore,
`x +1/x = 4 - sqrt15 + 4 + sqrt15`
` = 4+ 4 `
= ` 8 `
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