`sqrt(x) + sqrt(x-8) = 0`

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You need to solve for x the following equation, such that:

sqrt x + sqrt (x - 8) = 0

You need to raise to square both sides, using the following formula, such that:

(a + b)^2 = a^2 + 2ab + b^2

Considering a = sqrt x, b = sqrt(x - 8) yields:

(sqrt x + sqrt (x - 8))^2 = x + 2sqrt(x(x - 8)) + x - 8 => (sqrt x + sqrt (x - 8))^2 = 0 => 2x - 8 + 2sqrt(x(x - 8)) = 0

Isolating the square root to the left side, yields:

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

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

Raising to square again, yields:

(x(x - 8)) = (4 - x)^2 => x^2 - 8x = 16 - 8x + x^2

Reducing duplicate terms both sides yields:

0 = 16 invalid

Hence, evaluating the solution to the given equation yields that there are no solutions.

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