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Question
In the following, find the value of a and b:
`(sqrt(3) - 1)/(sqrt(3) + 1) + (sqrt(3) + 1)/(sqrt(3) - 1) = "a" + "b"sqrt(3)`
Solution
`(sqrt(3) - 1)/(sqrt(3) + 1) + (sqrt(3) + 1)/(sqrt(3) - 1)`
= `(sqrt(3) - 1)/(sqrt(3) + 1) xx (sqrt(3) - 1)/(sqrt(3) - 1) + (sqrt(3) + 1)/(sqrt(3) - 1) + (sqrt(3) + 1)/(sqrt(3) + 1)`
= `(sqrt(3) - 1)^2/((sqrt(3))^2 - 1) + (sqrt(3) + 1)^2/((sqrt(3))^2 - 1)`
= `((sqrt(3))^2 - 2 xx sqrt(3) xx 1 + 1^2)/(3 - 1) + ((sqrt(3))^2 + 2 xx sqrt(3) xx 1 + 1^2)/(3 - 1)`
= `(3 - 2sqrt(3) + 1)/(2) + (3 + 2sqrt(3) + 1)/(2)`
= `(4 - 2sqrt(3))/(2) + (4 + 2sqrt(3))/(2)`
= `(2(2 - sqrt(3)))/(2) + (2(2 + sqrt(3)))/(2)`
= `2 - sqrt(3) + 2 + sqrt(3)`
= 4 + 0
Hence, a = 4 and b = 0
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