Question

Find x, if `16((a - x)/(a + x))^3 = (a + x)/(a - x)`.

Answer 1

`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`