Let's simplify the equation step by step:
√x - 2 - √2x + 2 = √2x - 5 - √3x - 1
Combine like terms:
√x - √2x - √3x = 2 - 2 - 5 - 1
Simplify the square roots:
√x(1 - 2 - √3) = -6
Now, we need to isolate the square root using algebraic operations:
√x(-1 - √3) = -6
Divide both sides by (-1 - √3):
√x = -6 / (-1 - √3)
Rationalize the denominator by multiplying by the conjugate of the denominator:
√x = (-6(-1 + √3)) / ((-1 - √3)(-1 + √3))
√x = (6 - 6√3) / (1 - 3)
√x = (6 - 6√3) / -2
√x = -3 + 3√3
Square both sides to solve for x:
x = (-3 + 3√3)^2
x = 9 - 18√3 + 27
x = 36 - 18√3
Therefore, the solution to the equation √x - 2 - √2x + 2 = √2x - 5 - √3x - 1 is x = 36 - 18√3.
Let's simplify the equation step by step:
√x - 2 - √2x + 2 = √2x - 5 - √3x - 1
Combine like terms:
√x - √2x - √3x = 2 - 2 - 5 - 1
Simplify the square roots:
√x(1 - 2 - √3) = -6
Now, we need to isolate the square root using algebraic operations:
√x(-1 - √3) = -6
Divide both sides by (-1 - √3):
√x = -6 / (-1 - √3)
Rationalize the denominator by multiplying by the conjugate of the denominator:
√x = (-6(-1 + √3)) / ((-1 - √3)(-1 + √3))
√x = (6 - 6√3) / (1 - 3)
√x = (6 - 6√3) / -2
√x = -3 + 3√3
Square both sides to solve for x:
x = (-3 + 3√3)^2
x = 9 - 18√3 + 27
x = 36 - 18√3
Therefore, the solution to the equation √x - 2 - √2x + 2 = √2x - 5 - √3x - 1 is x = 36 - 18√3.