`3 - (3 + sqrt5)/(4) - (1)/(3 + sqrt5)` is equal to

Options

A. 0
B. `3/2`
C. `sqrt5/2`
D. `sqrt5`

Answer

To solve the expression \( 3 - \frac{3 + \sqrt{5}}{4} - \frac{1}{3 + \sqrt{5}} \), we will follow these steps:

### Step 1: Simplify the expression
We start with the expression:
\[ 3 - \frac{3 + \sqrt{5}}{4} - \frac{1}{3 + \sqrt{5}} \]

### Step 2: Combine the fractions
To combine the fractions, we need a common denominator. The common denominator for \( 4 \) and \( 3 + \sqrt{5} \) is \( 4(3 + \sqrt{5}) \). We rewrite each term with this common denominator:
\[ 3 = \frac{3 \cdot 4(3 + \sqrt{5})}{4(3 + \sqrt{5})} = \frac{12(3 + \sqrt{5})}{4(3 + \sqrt{5})} \]
\[ -\frac{3 + \sqrt{5}}{4} = -\frac{(3 + \sqrt{5})(3 + \sqrt{5})}{4(3 + \sqrt{5})} = -\frac{(3 + \sqrt{5})^2}{4(3 + \sqrt{5})} \]
\[ -\frac{1}{3 + \sqrt{5}} = -\frac{4}{4(3 + \sqrt{5})} \]

### Step 3: Combine all terms
Now we can combine all the terms over the common denominator:
\[ \frac{12(3 + \sqrt{5}) - (3 + \sqrt{5})^2 - 4}{4(3 + \sqrt{5})} \]

### Step 4: Expand and simplify the numerator
First, we need to expand \( (3 + \sqrt{5})^2 \):
\[ (3 + \sqrt{5})^2 = 9 + 6\sqrt{5} + 5 = 14 + 6\sqrt{5} \]
Now substitute this back into the numerator:
\[ 12(3 + \sqrt{5}) - (14 + 6\sqrt{5}) - 4 \]
Expanding \( 12(3 + \sqrt{5}) \):
\[ = 36 + 12\sqrt{5} \]
Now combine:
\[ 36 + 12\sqrt{5} - 14 - 6\sqrt{5} - 4 = 36 - 14 - 4 + (12\sqrt{5} - 6\sqrt{5}) = 18 + 6\sqrt{5} \]

### Step 5: Final expression
Now we have:
\[ \frac{18 + 6\sqrt{5}}{4(3 + \sqrt{5})} \]
We can simplify this further:
\[ = \frac{6(3 + \sqrt{5})}{4(3 + \sqrt{5})} = \frac{6}{4} = \frac{3}{2} \]

### Final Answer
Thus, the value of the expression \( 3 - \frac{3 + \sqrt{5}}{4} - \frac{1}{3 + \sqrt{5}} \) is:
\[ \frac{3}{2} \]