Question 124675
{{{sqrt(x-1)+sqrt(x-4)=sqrt(2x-1)}}} Start with the given equation



{{{(sqrt(x-1)+sqrt(x-4))^2=(sqrt(2x-1))^2}}} Square both sides



{{{(sqrt(x-1)+sqrt(x-4))^2=2x-1}}} Square the right side to remove the square root



{{{sqrt(x-1)sqrt(x-1)+2sqrt(x-1)sqrt(x-4)+sqrt(x-4)sqrt(x-4)=2x-1}}} Foil the left side


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



{{{(x-1)+2sqrt(x^2-5x+4)+(x-4)=2x-1}}} Foil



{{{2x-5+2sqrt(x^2-5x+4)=2x-1}}} Combine like terms



{{{2sqrt(x^2-5x+4)=2x-1-2x+5}}} Subtract 2x from both sides. Add 5 to both sides.



{{{2sqrt(x^2-5x+4)=4}}} Combine like terms



{{{sqrt(x^2-5x+4)=2}}} Divide both sides by 2



{{{x^2-5x+4=4}}} Square both sides



{{{x^2-5x=0}}}  Subtract 4 from both sides. 



{{{x(x-5)=0}}} Factor the left side 



Now set each factor equal to zero:

{{{x=0}}} or  {{{x-5=0}}} 


{{{x=0}}} or  {{{x=5}}}    Now solve for x in each case



So our possible answers are:

 {{{x=0}}} or  {{{x=5}}} 




Check:

Let's verify the first solution {{{x=0}}}


{{{sqrt(x-1)+sqrt(x-4)=sqrt(2x-1)}}} Start with the given equation



{{{sqrt(0-1)+sqrt(0-4)=sqrt(2(0)-1)}}} Plug in {{{x=0}}} 



{{{sqrt(-1)+sqrt(-4)=sqrt(-1)}}} Combine like terms



{{{i+2i=i}}} Simplify



{{{3i=i}}} Add.  Since the two sides of the equation are <b>not</b> equal, this means that {{{x=0}}} is <b>not</b> a solution.



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Let's verify the second solution {{{x=5}}}


{{{sqrt(x-1)+sqrt(x-4)=sqrt(2x-1)}}} Start with the given equation



{{{sqrt(5-1)+sqrt(5-4)=sqrt(2(5)-1)}}} Plug in {{{x=5}}} 



{{{sqrt(4)+sqrt(1)=sqrt(9)}}} Combine like terms



{{{2+1=3}}} Take the square root



{{{3=3}}} Add.  Since the two sides of the equation are equal, this verifies the solution



So the only solution is {{{x=5}}}