2012/026) Solve: (sqrt(1+3x)-sqrt(2x-1)) = sqrt(x+2)

given
(sqrt(1+3x)-sqrt(2x-1)) = sqrt(x+2) ..1

we know
(1+3x) - (2x-1) = (x+ 2)

so given
(sqrt(1+3x)-sqrt(2x-1))(sqrt(1+3x) +sqrt(2x-1)) = (x+2) ..2

divide (2) by (1) but before dividing check that x + 2 is zero is not a solution

x = 0 => x = - 2

putting in (1) we see the solution is satisfied.

Assume x is not 2 and we devide to get

(sqrt(1+3x) +sqrt(2x-1)) = sqrt(x+2) ..3

from (1) and (3)

(sqrt(1+3x) +sqrt(2x-1)) = (sqrt(1+3x) -sqrt(2x-1))

so sqrt(2x-1) = 0 or x = 1/2

check:
LHS = (sqrt(1+3x)-sqrt(2x-1)) = (sqrt(5/2)-sqrt(0))= sqrt(5/2))
RHS = sqrt(5/2)

x = {1/2 ,-2} is the solution set