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Question
If x = `((2 + sqrt(5)))/((2 - sqrt(5))` and y = `((2 - sqrt(5)))/((2 + sqrt(5))`, show that (x2 - y2) = `144sqrt(5)`.
Solution
x = `((2 + sqrt(5)))/((2 - sqrt(5))`
= `((2 + sqrt(5)))/((2 - sqrt(5))) xx ((2 + sqrt(5)))/((2 + sqrt(5))`
= `(2 + sqrt(5))^2/(4 - 5)`
= `-(4 + 5 + 4sqrt(5))`
= `-9 -4sqrt(5)`
y = `((2 - sqrt(5)))/((2 + sqrt(5))`
= `((2 - sqrt(5)))/((2 + sqrt(5))) xx ((2 - sqrt(5)))/((2 - sqrt(5))`
= `(2 - sqrt(5))^2/(4 - 5)`
= `-(4 + 5 -4sqrt(5))`
= `-9 + 4sqrt(5)`
∴ x2 - y2 = (x + y) (x - y)
= `(-9 - 4sqrt(5) - 9 + 4sqrt(5))(-9 -4sqrt(5) + 9 - 4sqrt(5))`
= `(-18)(-8sqrt(5))`
= `144sqrt(5)`
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