Question

If `x = (sqrt(m + n) + sqrt(m - n))/(sqrt(m + n) - sqrt(m - n))`, express n in terms of x and m.

Answer 1

`x = (sqrt(m + n) + sqrt(m - n))/(sqrt(m + n) - sqrt(m - n)`

Applying componendo and dividendo,

`(x + 1)/(x - 1) = (sqrt(m + n) + sqrt(m - n) + sqrt(m + n) - sqrt(m - n))/(sqrt(m + n) + sqrt(m - n) - sqrt(m + n) + sqrt(m - n))`

`(x + 1)/(x - 1) =  (2sqrt(m + n))/(2sqrt(m - n))`

Squaring both sides,

`(x^2 + 2x + 1)/(x^2 - 2x + 1) = (m + n)/(m - n)`

Applying componendo and dividendo,

`(x^2 + 2x + 1 + x^2 - 2x + 1 )/(x^2 + 2x  + 1 - x^2 + 2x - 1) = 
(m + n + m - n)/(m + n - m + n)`

`(2x^2 + 2)/(4x) = (2m)/(2n)`

`(x^2 + 1)/(2x) = m/n`

`(x^2 + 1)/(2mx) = 1/n`

`n = (2mx)/(x^2 + 1)`