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Question
Find x, if `16((a - x)/(a + x))^3 = (a + x)/(a - x)`.
Solution
`16((a - x)/(a + x))^3 = (a + x)/(a - x)`
`\implies ((a + x)/(a - x)) xx ((a + x)/(a - x))^3 = 16`
`\implies ((a + x)/(a - x))^4 = 16 = (±2)^4`
`\implies (a + x)/(a - x) = ± 2`
When `(a + x)/(a - x) = 2/1`
Applying componendo and dividendo,
`(a + x + a - x)/(a + x - a + x) = (2 + 1)/(2 - 1)`
`\implies (2a)/(2x) = 3/1`
`\implies a/x = 3/1`
`\implies` 3x = a
∴ `x = a/3`
When `(a + x)/(a - x) = (-2)/1`
Applying componendo and dividendo,
`(a + x + a - x)/(a + x - a + x) = (-2 + 1)/(-2 - 1)`
`\implies (2a)/(2x) = (-1)/(-3)`
`\implies a/x = 1/3`
`\implies` x = 3a
Hence `x = a/3, 3a`
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