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Question
Using componendo and dividendo, find the value of x:
`(sqrt(3x + 4) + sqrt(3x - 5))/(sqrt(3x + 4) - sqrt(3x - 5)) = 9`
Solution
`(sqrt(3x + 4) + sqrt(3x - 5))/(sqrt(3x + 4) - sqrt(3x - 5)) = 9/1`
Applying componendo and dividend, we have
`(sqrt(3x + 4) + sqrt(3x - 5) + sqrt(3x + 4) - sqrt(3x - 5))/(sqrt(3x + 4) + sqrt(3x - 5) - sqrt(3x + 4) + sqrt(3x - 5)) = (9 + 1)/(9 - 1)`
`=> (2sqrt(3x + 4))/(2sqrt(3x - 5)) = 5/4`
Squaring both sides, we have
`(3x + 4)/(3x - 5) = 25/16`
`=>` 16(3x + 4) = 25(3x – 5)
`=>` 48x – 64 = 75x – 125
`=>` 75x – 48x = 64 + 125
`=>` 27x = 189
`=> x = 189/27`
`=>` x = 7
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