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Question
Solve x : `(sqrt(36x + 1) + 6sqrt(x))/(sqrt(36x + 1) -6sqrt(x))` = 9
Solution
`(sqrt(36x + 1) + 6sqrt(x))/(sqrt(36x + 1) -6sqrt(x)) = (9)/(1)`
Applying componendo and dividendo,
`(sqrt(36x + 1) + 6sqrt(x) + sqrt(36x + 1) - 6sqrt(x))/(sqrt(36x + 1) + 6sqrt(x) - sqrt(36x - 1) + 6sqrt(x)`
= `(9 + 1)/(9 - 1)`
⇒ `(2sqrt(36x + 1))/(12sqrt(x)) = (10)/(8)`
⇒ `sqrt(36x + 1)/(6sqrt(x)) = (5)/(4)` ...(Squaring both sides)
`(36x + 1)/(36x) = (25)/(16)`
⇒ 36x x 25 = 16(36x + 1)
⇒ 900x = 576x + 16
⇒ 900x - 576x = 16
⇒ 324 = 16
∴ x = `(16)/(324)`
= `(4)/(81)`.
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