Question

Using properties of proportion, solve for x. Given that x is positive:

`(2x + sqrt(4x^2 -1))/(2x - sqrt(4x^2 - 1)) = 4`

Using properties of proportion, find the value of x from the following:

`(2x + sqrt(4x^2 -1))/(2x - sqrt(4x^2 - 1)) = 4`

Answer 1

`(2x + sqrt(4x^2 -1))/(2x - sqrt(4x^2 - 1)) = 4`

⇒ `(2x + sqrt(4x^2 -1) + 2x  - sqrt(4x^2 - 1))/(2x + sqrt(4x^2 - 1) - 2x +  sqrt(4x^2 - 1)) = (4+1)/(4-1) `     (By componendo dividendo)

⇒ `(4x)/(2sqrt(4x^2 -1)) = 5/3`

⇒ `(2x)/(sqrt(4x^2 -1)) = 5/3`

Squaring both sides,

⇒ `((2x)/sqrt(4x^2 - 1))^2 = (5/3)^2`

⇒ `(4x^2)/(4x^2 -1) = 25/9`  

⇒ 9(4x²) = 25(4x² − 1)

⇒ 36x² = 100x² − 25

⇒ 64x² = 25

⇒ x² = `25/64`

⇒ x = `5/8`