To determine the quotient of [tex]\(\frac{1}{1 + \sqrt{3}}\)[/tex], we need to rationalize the denominator, a process that eliminates the square root from the denominator.
### Step 1: Rationalize the denominator
To rationalize the denominator of [tex]\(\frac{1}{1 + \sqrt{3}}\)[/tex], multiply both the numerator and the denominator by the conjugate of the denominator, [tex]\(1 - \sqrt{3}\)[/tex]. The conjugate is used because it can simplify the expressions involving square roots.
[tex]\[
\frac{1}{1 + \sqrt{3}} \cdot \frac{1 - \sqrt{3}}{1 - \sqrt{3}} = \frac{1 \cdot (1 - \sqrt{3})}{(1 + \sqrt{3}) \cdot (1 - \sqrt{3})}
\][/tex]
### Step 2: Simplify the numerator
The numerator becomes:
[tex]\[
1 \cdot (1 - \sqrt{3}) = 1 - \sqrt{3}
\][/tex]
### Step 3: Simplify the denominator
Use the difference of squares formula for the denominator:
[tex]\[
(1 + \sqrt{3})(1 - \sqrt{3}) = 1^2 - (\sqrt{3})^2 = 1 - 3 = -2
\][/tex]
So, the expression becomes:
[tex]\[
\frac{1 - \sqrt{3}}{-2}
\][/tex]
### Step 4: Remove the negative from the denominator
Simplify by changing the signs in the numerator:
[tex]\[
\frac{1 - \sqrt{3}}{-2} = \frac{-(1 - \sqrt{3})}{2} = \frac{-1 + \sqrt{3}}{2}
\][/tex]
### Step 5: Compare to given choices
The answer matches one of the given choices. Among the choices given:
1. [tex]\(\frac{\sqrt{3}}{4}\)[/tex]
2. [tex]\(\frac{1 + \sqrt{3}}{4}\)[/tex]
3. [tex]\(\frac{1 - \sqrt{3}}{4}\)[/tex]
4. [tex]\(\frac{-1 + \sqrt{3}}{2}\)[/tex]
The correct choice is:
[tex]\[
\boxed{\frac{-1 + \sqrt{3}}{2}}
\][/tex]
