Question

Find x from the following equations : `(3x + sqrt(9x^2 - 5))/(3x - sqrt(9x^2 - 5)) = (5)/(1)`

Answer 1

`(3x + sqrt(9x^2 - 5))/(3x - sqrt(9x^2 - 5)) = (5)/(1)`
Applying 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)/(2sqrt(9x^2 - 5)) = (6)/(4)`

⇒ `(3x)/sqrt(9x^2 - 5) = (3)/(2)`
Squaring both sides
`(9x^2)/(9x^2 - 5) = (9)/(4)`

⇒ 81x² – 45 = 36x²
⇒ 81x² – 36x² = 45
⇒ 45x² = 45
⇒ x² = 1
⇒ x = ± 1
∴ x = 1, –1
Check :
(i) When x = 1, then in the given equation
`(3 xx 1 + sqrt(9 xx 1 - 5))/(3 xx 1 - sqrt(9 xx 1 - 5)`

= `(3 + sqrt(4))/( 3 - sqrt4)` 

= `(3 + 2)/(3 - 2)`

= `(5)/(1)`
which is given
∴ x = 1
(ii) When x = –1, then

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

= `(-3 + sqrt(9 - 5))/(-3 - sqrt(9 - 5)`

= `(-3 + sqrt(4))/(-3 - sqrt(4)`

= `(-3 + 2)/(-3 - 2)`

= `(-1)/(-5)`

= `(1)/(5) ≠ (5)/(1)`
∴ x = –1 is not its solution.
Hence x = 1.