Question 916040
{{{sqrt(2x+3)-sqrt(x+1)=1}}}


1.  Isolate one of the square roots by subtracting the other from both sides.  In this case, I will add {{{-sqrt(x+1)}}} from both sides, giving us


{{{sqrt(2x+3)=1+sqrt(x+1)}}}


2.  Square both sides so that we eliminate the root on the left side.  This will give us


{{{2x+3=1+2*sqrt(x+1)+x+1}}}


3.  Combine like terms on the right side, giving us


{{{2x+3=2+2*sqrt(x+1)+x}}}


4.  Subtract the 2 on the right side from both sides of the equal sign, which gives us


{{{2x+3-2=2*sqrt(x+1)+x}}}----->{{{2x+1=2*sqrt(x+1)+x}}}


5.  Subtract the x on the right side from both sides of the equal sign, which gives us


{{{2x+1-x=2*sqrt(x+1)}}}----->{{{x+1=2*sqrt(x+1)}}}


6.  Divide both sides of the equation by 2, giving us


{{{(x+1)/2=sqrt(x+1)}}}


7.  Square both sides of the equal sign to eliminate the radical on the right side, giving us


{{{(x^2+2x+1)/4=x+1}}}


8.  Multiply both sides of the equal sign by 4 to get rid of the fraction on the left side, giving us:


{{{x^2+2x+1=4x+4}}}


9.  Place all terms on the left side of the equation and set the equation equal to zero:


{{{x^2+2x-4x+1-4=0}}}----->{{{x^2-2x-3=0}}}


10.  Factor the left side of the equation, which will give us


{{{(x+1)(x-3)=0}}}


11.  Set each factor equal to 0 and solve for x for both factors:


{{{x+1=0}}}and{{{x-3=0}}}----->{{{x=-1}}} and {{{x=3}}}


12.  Verify that both of these values of x work, by plugging them into the original equation and making sure when plugged in, they equal 1.


Final Answer:  x = -1 , 3