What is the square root of x + the square root of x - 8 = 2?

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Solve `sqrt(x)+sqrt(x-8)=2`

Square both sides (Note that this might introduce extraneous solutions)

`(sqrt(x)+sqrt(x-8))^2=2^2`

`x+2sqrt(x)sqrt(x-8)+x-8=4`

`2sqrt(x)sqrt(x-8)=-2x+12`

`sqrt(x)sqrt(x-8)=-x+6`

Again square both sides:

`x(x-8)=x^2-12x+36`

`x^2-8x=x^2-12x+36`

4x=36

x=9

Since we squared both sides of the equation, we must check if the purported solution is extraneous by substituting into the original equation:

`sqrt(9)+sqrt(9-8)=3+1=4 !=2` so the solution is extraneous.

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**There is no solution to the equation**

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The graph of `y=sqrt(x)+sqrt(x-8)` and the graph of y=2:

Note that the graphs do not intersect.

`sqrt(x) + sqrt(x-8) = 2`

subtract both dìsides `sqrt(x)`

`sqrt(x) + sqrt(x-8) -sqrt(x) = 2 - sqrt(x)`

`sqrt(x- 8)= 2-sqrt(x)`

on the square:

`x-8=4 -4sqrt(x) +x`

subtrcting x both sides:

`x-8 -x = 4 - 4sqrt(x) + x -x```

`-8=4 - 4sqrt(x)`

subtracting 4 both sides:

`-8 -4 = 4 - 4sqrt(x) -4`

`-12= -4sqrt(x)`

dividing both sdes by - 4:

`-12/-4 = -4/-4 sqrt(x)`

`3= sqrt(x)` `x=9`

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