To find which choice is equivalent to the given fraction [tex]\(\frac{6}{6+\sqrt{12x}}\)[/tex] when [tex]\(x\)[/tex] is an appropriate value, we need to rationalize the denominator and simplify. Here’s the step-by-step solution:
1. Identify the given fraction:
[tex]\[
\frac{6}{6+\sqrt{12x}}
\][/tex]
2. Rationalize the denominator:
To rationalize the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\(6 + \sqrt{12x}\)[/tex] is [tex]\(6 - \sqrt{12x}\)[/tex].
[tex]\[
\frac{6}{6+\sqrt{12x}} \times \frac{6-\sqrt{12x}}{6-\sqrt{12x}}
\][/tex]
3. Multiply the numerator:
[tex]\[
6 \cdot (6 - \sqrt{12x}) = 6 \cdot 6 - 6 \cdot \sqrt{12x}
\][/tex]
Simplify this expression:
[tex]\[
36 - 6\sqrt{12x}
\][/tex]
4. Multiply the denominator:
Use the difference of squares formula [tex]\((a+b)(a-b) = a^2 - b^2\)[/tex]:
[tex]\[
(6 + \sqrt{12x})(6 - \sqrt{12x}) = 6^2 - (\sqrt{12x})^2
\][/tex]
Simplify this expression:
[tex]\[
36 - 12x
\][/tex]
5. Rewrite the fraction:
Now we have:
[tex]\[
\frac{36 - 6\sqrt{12x}}{36 - 12x}
\][/tex]
6. Simplify the expression:
We observe that the expressions can be simplified further if needed, but for comparison purposes, we will look at the given options.
After these steps, the fraction becomes:
[tex]\[
\frac{36 - 6\sqrt{12x}}{36 - 12x}
\][/tex]
Next, we compare the result with the given choices:
- A. [tex]\(\frac{6 - \sqrt{12x}}{36 - 12x}\)[/tex]
- B. [tex]\(\frac{6 - \sqrt{12x}}{6 - 2x}\)[/tex]
- C. [tex]\(\frac{6 - \sqrt{12x}}{36 - 6x}\)[/tex]
- D. [tex]\(\frac{6 - \sqrt{12x}}{6 - 12x}\)[/tex]
The simplified numerator and denominator closely match the given choice A if reduced, indicating they represent the same ratio:
[tex]\(
36 - 6\sqrt{12x} = 6(6 - \sqrt{12x})
\)[/tex]
Thus, the simplified form of the fraction matches choice:
[tex]\[
A. \frac{6 - \sqrt{12x}}{36 - 12x}
\][/tex]
Therefore, the equivalent choice for the given fraction [tex]\(\frac{6}{6+\sqrt{12x}}\)[/tex] is:
[tex]\[
\boxed{\frac{6 - \sqrt{12x}}{36 - 12x}}
\][/tex]
