To determine which choice is equivalent to the given fraction [tex]\(\frac{2}{2 - \sqrt{6x}}\)[/tex], we need to rationalize the denominator and simplify it.
The expression given is:
[tex]\[
\frac{2}{2 - \sqrt{6x}}
\][/tex]
To rationalize the denominator, we will multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\(2 - \sqrt{6x}\)[/tex] is [tex]\(2 + \sqrt{6x}\)[/tex]. Multiplying by this conjugate will help to eliminate the square root in the denominator.
So, we multiply the fraction by [tex]\(\frac{2 + \sqrt{6x}}{2 + \sqrt{6x}}\)[/tex]:
[tex]\[
\frac{2}{2 - \sqrt{6x}} \cdot \frac{2 + \sqrt{6x}}{2 + \sqrt{6x}} = \frac{2(2 + \sqrt{6x})}{(2 - \sqrt{6x})(2 + \sqrt{6x})}
\][/tex]
First, let's simplify the numerator:
[tex]\[
2(2 + \sqrt{6x}) = 4 + 2\sqrt{6x}
\][/tex]
Next, we simplify the denominator. The denominator is a difference of squares:
[tex]\[
(2 - \sqrt{6x})(2 + \sqrt{6x}) = 2^2 - (\sqrt{6x})^2 = 4 - 6x
\][/tex]
Now, putting it all together, we have:
[tex]\[
\frac{4 + 2\sqrt{6x}}{4 - 6x}
\][/tex]
Thus, the rationalized and simplified form of the given fraction [tex]\(\frac{2}{2 - \sqrt{6x}}\)[/tex] is:
[tex]\[
\frac{4 + 2\sqrt{6x}}{4 - 6x}
\][/tex]
To match one of the answer choices, we can simplify by dividing both the numerator and the denominator by 2:
[tex]\[
\frac{4 + 2\sqrt{6x}}{4 - 6x} = \frac{2(2 + \sqrt{6x})}{2(2 - 3x)} = \frac{2 + \sqrt{6x}}{2 - 3x}
\][/tex]
Therefore, the correct choice is:
[tex]\[
\boxed{\frac{2 + \sqrt{6x}}{4 - 6x}}
\][/tex]
So, the correct answer is:
D. [tex]\(\frac{2 + \sqrt{6x}}{4 - 6x}\)[/tex]
