Question 65135
Solve for x: sqrt(2x-1) - sqrt(x+3) = 1
{{{sqrt(2x-1)-sqrt(x+3)+sqrt(x+3)=sqrt(x+3)+1}}}
{{{sqrt(2x-1)=sqrt(x+3)+1}}}
{{{(sqrt(2x-1))^2=(sqrt(x+3)+1)^2}}}
{{{2x-1=sqrt((x+3)^2)+2sqrt(x+3)+1}}}
{{{2x-1=x+3+2sqrt(x+3)+1}}}
{{{2x-1=x+4+2sqrt(x+3)}}}
{{{2x-x-1-4=x-x+4-4+2sqrt(x+3)}}}
{{{x-5=2sqrt(x+3)}}}
{{{(x-5)^2=(2sqrt(x+3))^2}}}
{{{x^2-10x+25=4(x+3)}}}
{{{x^2-10x+25=4x+12}}}
{{{x^2-10x-4x+25-12=4x-4x+12-12}}}
{{{x^2-14x+13=0}}}
(x-1)(x-13)=0
x-1=0  or x-13=0
x-1+1=0+1 or x-13+13=0+13
x=1 or x=13  Check for extraneous solutions.
{{{sqrt(2x-1)-sqrt(x+3)=1}}}
{{{sqrt(2(1)-1)-sqrt(1+3)=1}}}
{{{sqrt(2-1)-sqrt(4)=1}}}
{{{sqrt(1)-sqrt(4)=1}}}
{{{1-2=1}}}
{{{-1=1}}}  False! x=1 is extraneous!
{{{sqrt(2(13)-1)-sqrt(13+3)=1}}}
{{{sqrt(26-1)-sqrt(16)=1}}}
{{{sqrt(25)-sqrt(16)=1}}}
{{{5-4=1}}}
{{{1=1}}}  True! x=13 is a valid solution!!!
Happy Calculating!!!