To determine which choice is equivalent to the given fraction [tex]\(\frac{4}{\sqrt{x-2}-\sqrt{x}}\)[/tex], we first need to rationalize the denominator and then simplify the expression.
### Step-by-Step Solution:
1. Given Expression:
[tex]\[
\frac{4}{\sqrt{x-2} - \sqrt{x}}
\][/tex]
2. Rationalize the Denominator:
To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator [tex]\(\sqrt{x-2} + \sqrt{x}\)[/tex]:
[tex]\[
\frac{4}{\sqrt{x-2} - \sqrt{x}} \times \frac{\sqrt{x-2} + \sqrt{x}}{\sqrt{x-2} + \sqrt{x}}
\][/tex]
3. Multiply the Numerator and Denominator:
The numerator becomes:
[tex]\[
4 \times (\sqrt{x-2} + \sqrt{x})
\][/tex]
The denominator is a 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]
4. Form the New Fraction:
The expression now simplifies to:
[tex]\[
\frac{4(\sqrt{x-2} + \sqrt{x})}{-2}
\][/tex]
5. Simplifying Further:
Divide the numerator by [tex]\(-2\)[/tex]:
[tex]\[
\frac{4}{-2} \times (\sqrt{x-2} + \sqrt{x}) = -2(\sqrt{x-2} + \sqrt{x})
\][/tex]
6. Rewriting Simplified Expression:
[tex]\[
-2(\sqrt{x} + \sqrt{x-2})
\][/tex]
### Conclusion:
The simplified expression is:
[tex]\[
-2(\sqrt{x} + \sqrt{x-2})
\][/tex]
Therefore, the correct choice is:
Answer: B. [tex]\(-2(\sqrt{x} + \sqrt{x-2})\)[/tex]
