Answer :
Let's solve the problem step-by-step.
We start with the given fraction:
[tex]\[ \frac{4}{\sqrt{x-2} - \sqrt{x}} \][/tex]
To simplify this fraction, we rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\(\sqrt{x-2} - \sqrt{x}\)[/tex] is [tex]\(\sqrt{x-2} + \sqrt{x}\)[/tex].
Thus, we proceed as follows:
[tex]\[ \frac{4}{\sqrt{x-2} - \sqrt{x}} \times \frac{\sqrt{x-2} + \sqrt{x}}{\sqrt{x-2} + \sqrt{x}} = \frac{4 (\sqrt{x-2} + \sqrt{x})}{(\sqrt{x-2} - \sqrt{x})(\sqrt{x-2} + \sqrt{x})} \][/tex]
Next, we simplify the denominator using the difference of squares:
[tex]\[ (\sqrt{x-2} - \sqrt{x})(\sqrt{x-2} + \sqrt{x}) = (\sqrt{x-2})^2 - (\sqrt{x})^2 = (x-2) - x = -2 \][/tex]
So, the fraction becomes:
[tex]\[ \frac{4 (\sqrt{x-2} + \sqrt{x})}{-2} \][/tex]
This simplifies by dividing both the numerator and the denominator by [tex]\(-2\)[/tex]:
[tex]\[ \frac{4 (\sqrt{x-2} + \sqrt{x})}{-2} = -2 (\sqrt{x-2} + \sqrt{x}) \][/tex]
Now we compare this result with the given choices:
A. [tex]\(-2 (\sqrt{x} - \sqrt{x-2})\)[/tex]
B. [tex]\(2 (\sqrt{x} + \sqrt{x-2})\)[/tex]
C. [tex]\(-2 (\sqrt{x} + \sqrt{x-2})\)[/tex]
D. [tex]\(2 (\sqrt{x} - \sqrt{x-2})\)[/tex]
From our simplification, we have [tex]\(-2 (\sqrt{x-2} + \sqrt{x})\)[/tex], which matches choice C.
Therefore, the correct answer is:
[tex]\[ \boxed{C} \][/tex]
We start with the given fraction:
[tex]\[ \frac{4}{\sqrt{x-2} - \sqrt{x}} \][/tex]
To simplify this fraction, we rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\(\sqrt{x-2} - \sqrt{x}\)[/tex] is [tex]\(\sqrt{x-2} + \sqrt{x}\)[/tex].
Thus, we proceed as follows:
[tex]\[ \frac{4}{\sqrt{x-2} - \sqrt{x}} \times \frac{\sqrt{x-2} + \sqrt{x}}{\sqrt{x-2} + \sqrt{x}} = \frac{4 (\sqrt{x-2} + \sqrt{x})}{(\sqrt{x-2} - \sqrt{x})(\sqrt{x-2} + \sqrt{x})} \][/tex]
Next, we simplify the denominator using the difference of squares:
[tex]\[ (\sqrt{x-2} - \sqrt{x})(\sqrt{x-2} + \sqrt{x}) = (\sqrt{x-2})^2 - (\sqrt{x})^2 = (x-2) - x = -2 \][/tex]
So, the fraction becomes:
[tex]\[ \frac{4 (\sqrt{x-2} + \sqrt{x})}{-2} \][/tex]
This simplifies by dividing both the numerator and the denominator by [tex]\(-2\)[/tex]:
[tex]\[ \frac{4 (\sqrt{x-2} + \sqrt{x})}{-2} = -2 (\sqrt{x-2} + \sqrt{x}) \][/tex]
Now we compare this result with the given choices:
A. [tex]\(-2 (\sqrt{x} - \sqrt{x-2})\)[/tex]
B. [tex]\(2 (\sqrt{x} + \sqrt{x-2})\)[/tex]
C. [tex]\(-2 (\sqrt{x} + \sqrt{x-2})\)[/tex]
D. [tex]\(2 (\sqrt{x} - \sqrt{x-2})\)[/tex]
From our simplification, we have [tex]\(-2 (\sqrt{x-2} + \sqrt{x})\)[/tex], which matches choice C.
Therefore, the correct answer is:
[tex]\[ \boxed{C} \][/tex]
