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Question
Find the value of a and b if `(sqrt(7) - 2)/(sqrt(7) + 2) = "a"sqrt(7) + "b"`
Solution
`(sqrt(7) - 2)/(sqrt(7) + 2) = "a"sqrt(7) + "b"`
⇒ `((sqrt(7) - 2)(sqrt(7) - 2))/((sqrt(7) + 2)(sqrt(7) - 2)) = "a"sqrt(7) + "b"`
⇒ `(sqrt(7) - 2)^2/((sqrt(7))^2 - 2^2) = "a"sqrt(7) + "b"`
`((sqrt(7))^2 - 2(sqrt(7))(2) + 2^2)/(7 - 4) = "a"sqrt(7) + "b"`
`(7 - 4sqrt(7) + 4)/3 = "a"sqrt(7) + "b"`
`(11 - 4sqrt(7))/3 = "a"sqrt(7) + "b"`
`11/3 + (-4 sqrt(7))/3 = "a"sqrt(7) + "b"`
∴ a = `- 4/3` and b = `11/3`
The value of a = `- 4/3` and b = `11/3`
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