Question

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

`(3x + sqrt(9x^2 - 5))/(3x - sqrt(9x^2 - 5))` = 5

Answer 1

We are given that

`(3x + sqrt(9x^2 - 5))/(3x - sqrt(9x^2 - 5)) = 5/1`

Apply Componendo and Dividendo,

⇒ `((3x + sqrt(9x^2 - 5)) + (3x - sqrt(9x^2 - 5)))/((3x + sqrt(9x^2 - 5)) - (3x - sqrt(9x^2 - 5))) = (5 + 1)/(5 - 1)`

⇒ `(6x)/(2 sqrt(9x^2 - 5)) = 6/4`

⇒ `(3x)/sqrt(9x^2 - 5) = 3/2`

⇒ `x/sqrt(9x^2 - 5) = 1/2`   ...(Dividing both sides by 3)

Square both sides

⇒ `(x/sqrt(9x^2 - 5))^2 = (1/2)^2`

⇒ `x^2/(9x^2 - 5) = 1/4`

⇒ 4x² = 9x² − 5

⇒ 5 = 5x²

⇒ x = ±1

But x = −1 does not satisfy the given equation. so, we reject x = −1.

Hence, x = 1.