Question

Solve the following equation.

`[sqrt( 4x + 1) + sqrt( x + 3 )]/[sqrt( 4x + 1 ) - sqrt( x+3 )]=4/1`

Answer 1

`[sqrt( 4x + 1) + sqrt( x + 3 )]/[sqrt( 4x + 1 ) - sqrt( x+3 )]=4/1`

Applying componendo and dividendo, we get

`{[sqrt( 4x + 1) + sqrt( x + 3 )] + [sqrt( 4x + 1 ) - sqrt( x+3 )]} /{[sqrt( 4x + 1) + sqrt( x + 3 )] - [sqrt( 4x + 1 ) - sqrt( x+3 )]}=(4+1)/(4-1)`

⇒ `(2sqrt[4x+1])/ (2sqrt[x + 3]) = 5/3`

⇒ `sqrt[4x+1]/sqrt[x + 3] = 5/3`

Squaring on both sides, we get

`(4x + 1)/(x +3) = (5/3)^2`

⇒ `(4x + 1)/(x +3) = 25/9`

⇒ `9(4x + 1) + 25(x +3)`

⇒ `36x +9 = 25x + 75`

⇒ `36x - 25x = 75 - 9`

⇒ `11x = 66`

⇒ `x = 6`

Thus, the solution of the given equation is x = 6.