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Answer 1 · 2015-05-07T19:36:46

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

Square both sides:
#(2x+1) -2sqrt((2x+1)(x+1))+ (x+1) = 4#

#3x+2 - 2 sqrt((2x+1)(x+1)) = 4#

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

Squaring both sides again:
#(2x+1)(x+1) = (9x^2-12x+4)/4#

#4(2x^2+3x+1) = 9x^2-12x+4#

#8x^2+12x+4 = 9x^2-12x+4#

#x^2-24x=0#

#x=0# or #x=24#

Testing #x=0# in #sqrt(2x+1)-sqrt(x+1) !=0#
so #x=0# is an extraneous solution.

Testing #x=24# in #sqrt(2x+1)-sqrt(x+1)#
#=sqrt(49) - sqrt(25)#
#=9-5#
#=2#

#x=24# is the valid solution.