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Question
If x =`[sqrt5 - 2 ]/[ sqrt5 + 2]` and y = `[ sqrt5 + 2]/[ sqrt5 - 2]`; find:
x2 + y2 + xy.
Sum
Solution
We rationalize the denominator,
`x = (sqrt5 - 2)/(sqrt5 + 2) xx (sqrt5 - 2)/(sqrt5 - 2)`
`x = (5 + 4 - 3sqrt5)/ (5 -4)`
`x = (9 - 4 sqrt5)/1`
Then, `x^2 = (9 - 4 sqrt5) (9 - 4 sqrt5)`
`= 81 + 16 ×5 - 72sqrt5 `
`= 161 - 72sqrt5`
We rationalize the denominator,
`y = (sqrt5 + 2)/(sqrt5 - 2) xx (sqrt5 + 2)/(sqrt5 + 2)`
`y = (5 + 4 + 4sqrt5)/(5-4)`
`y = (9 + 4sqrt5)/1`
Then, `y^2 = (9 + 4sqrt5) (9 + 4 sqrt5)`
`= 81 + 16 + 5 + 72 sqrt5`
`= 161 + 72sqrt5`
Now, `x = (sqrt5 + 2)/(sqrt5 - 2)`
and `y = (sqrt5 - 2)/(sqrt5 + 2)`
Then, `xy = (sqrt5 + 2)/(sqrt5 - 2) xx (sqrt5 - 2)/(sqrt5 + 2)`
xy = 1
Therefore, `x^2 + y^2 + xy`
`=161 - 72sqrt5 + 161 + 72sqrt5 + 1 = 323`
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Rationalisation of Surds
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