You can put this solution on YOUR website! You get to square everything and probably complete the square to solve.
first subtract sqrt(2x-1) from both sides.
Why? because you get that nice sqrt(x) all by itself on one side
sqrt(x) = 2- sqrt(2x-1)
now square the whole mess
(2- sqrt(2x-1))*(2- sqrt(2x-1))
use foil
first
2*2=4
Outside
2*(-sqrt(2x-1)
inside
2*(-sqrt(2x-1)
last
(-sqrt(2x-1))(-sqrt(2x-1))
2x-1
add them up and combine
x=4-2*(sqrt(2x-1)-2*(sqrt(2x-1)+2x-1
putting them in order
x=2x-4*(sqrt(2x-1))+3
If you think this is getting hairy we haven't begun the hairy part.
subtract x from both sides
0=x-4*(sqrt(2x-1))+3
now we want the x and 3 on e one side
so add 4*(sqrt(2x-1)) to both sides
4*(sqrt(2x-1))=x+3
divide both sides by 4
sqrt(2x-1)=(x+3)/4
square everything
2x-1=(x+3)^2/16
expand the terms on the right
2x-1=x^2/16+6x/16+9/16
believe it or not bring everything over to the left side to get ready for completing the square
2x-1-x^2/16-6x/16-9/16=0
combine terms
32x/16-16/16-x^2/16-6x/16-9/16=0
-x^2/16+13x/8-25/16=0
multiply by -16 or divide by -1/16
x^2-26x+25=0
subtract 25 from both sides.
complete the square
x^2-26x=-25
add (26/2)^2to both sides= 169
x^2-26x+169=144
factor
(x-13)=12
x=13+\-12
x=13+12
x=13-12
x=1
x=25
now we have to test both solutions to see if they fit in the original equation
sqrt(2x-1) + sqrt(x) = 2
let's check x=1
sqrt(2(1)-1)+sqrt(1)=2
sqrt(1)+sqrt(1)=2
1+1=2
yes
let's test 25
sqrt(2x-1) + sqrt(x) = 2
sqrt(50-1)+sqrt(25)=2
sqrt(49)+sqrt(25)=2
+\-7+\-5
+7-5=2
but the others
+7+5=12
-7-5=-12
-7+5=-2 don't work
