Question 143039
{{{sqrt(2x+1)-sqrt(x)=1}}} Start with the given equation



{{{sqrt(2x+1)=1+sqrt(x)}}} Add {{{sqrt(x)}}} to both sides 



{{{2x+1=(1+sqrt(x))^2}}} Square both sides 



{{{2x+1=1+2sqrt(x)+x}}} Foil the right side



{{{2x+1-1-x=2sqrt(x)}}} Subtract x from both sides. Subtract 1 from both sides



{{{x=2sqrt(x)}}} Combine like terms



{{{x^2=4x}}} Square both sides



{{{x^2-4x=0}}} Subtract 4x from both sides



{{{x(x-4)=0}}} Factor the left side




Now set each factor equal to zero:

{{{x=0}}} or  {{{x-4=0}}} 


{{{x=0}}} or  {{{x=4}}}    Now solve for x in each case



So our possible answers are


 {{{x=0}}} or  {{{x=4}}} 



However, we need to check our answers first



Check:

Let's check the solution {{{x=0}}} 



{{{sqrt(2x+1)-sqrt(x)=1}}} Start with the given equation



{{{sqrt(2(0)+1)-sqrt(0)=1}}} Plug in  {{{x=0}}} 



{{{sqrt(0+1)-sqrt(0)=1}}} Multiply 



{{{sqrt(1)-sqrt(0)=1}}} Add 



{{{1-0=1}}} Simplify the square roots



{{{1=1}}} Subtract. So this verifies the solution {{{x=0}}} 



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Now let's check the solution {{{x=4}}} 


{{{sqrt(2x+1)-sqrt(x)=1}}} Start with the given equation



{{{sqrt(2(4)+1)-sqrt(4)=1}}} Plug in  {{{x=4}}} 



{{{sqrt(8+1)-sqrt(4)=1}}} Multiply 



{{{sqrt(9)-sqrt(4)=1}}} Add 



{{{3-2=1}}} Simplify the square roots



{{{1=1}}} Subtract. So this verifies the solution {{{x=4}}} 





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Answer:


So the solutions are


 {{{x=0}}} or  {{{x=4}}}