Question

If `(x^(1/2) + x^(- 1/2))^2 = 9/2` then find the value of `(x^(1/2) - x^(-1/2))` for x >1

Answer 1

Given `(x^(1/2) + x^(- 1/2))^2 = 9/2` 

`(x^(1/2))^2 + (x^(-1/2))^2 + 2(x^(1/2))(x^(-1/2)) = 9/2`

`x^(1/2 xx 2) + x^(- 1/2 xx 2) + 2x^(1/2) xx 1/(x^(1/2)) = 9/2`

x¹ + x^–1 + 2 = `9/2`

`x + 1/x = 9/2 - 2`

`x + 1/x = (9 - 4)/2`

= `5/2`  ......(1)

Consider `(x^(1/2) - x^(-1/2))^2`

`(x^(1/2) - x^(-1/2))^2  = (x^(1/2))^2 + (x^(-1/2))^2 - 2x^(1/2) xx x^(-1/2)`

= `x^(1/2 xx 2) + x^(-1/2 xx 2) - 2x^(1/2) xx 1/(x^(1/2))`

= x¹ + x^–1 – 2

= `x + 1/x - 2`

= `5/2 - 2`

By equation (1)

= `(5 - 4)/2`

= `1/2`

∴ `x^(1/2) - x^(-1/2) = +-  sqrt(1/2)`

`x^(1/2) - x^(-1/2) = +-  1/sqrt(2)`

`x^(1/2) - x^(-1/2) = 1/sqrt(2)`

Since x > 1