Which of the following is equal to `(6 + sqrt(5))/(2 - sqrt(5))`?

Options

A. 17
B. `-17`
C. `17 + 8sqrt(5)`
D. `-(17 + 8sqrt(5))`

Answer

To simplify the expression \((6 + \sqrt{5})/(2 - \sqrt{5})\), we will follow these steps:

### Step 1: Multiply by the Conjugate
To simplify the expression, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(2 - \sqrt{5}\) is \(2 + \sqrt{5}\).

\[ \frac{6 + \sqrt{5}}{2 - \sqrt{5}} \cdot \frac{2 + \sqrt{5}}{2 + \sqrt{5}} = \frac{(6 + \sqrt{5})(2 + \sqrt{5})}{(2 - \sqrt{5})(2 + \sqrt{5})} \]

### Step 2: Simplify the Denominator
Now, we simplify the denominator using the difference of squares formula:

\[ (2 - \sqrt{5})(2 + \sqrt{5}) = 2^2 - (\sqrt{5})^2 = 4 - 5 = -1 \]

### Step 3: Expand the Numerator
Next, we expand the numerator:

\[ (6 + \sqrt{5})(2 + \sqrt{5}) = 6 \cdot 2 + 6 \cdot \sqrt{5} + \sqrt{5} \cdot 2 + \sqrt{5} \cdot \sqrt{5} \]

Calculating each term:

- \(6 \cdot 2 = 12\)
- \(6 \cdot \sqrt{5} = 6\sqrt{5}\)
- \(\sqrt{5} \cdot 2 = 2\sqrt{5}\)
- \(\sqrt{5} \cdot \sqrt{5} = 5\)

Adding these together:

\[ 12 + 6\sqrt{5} + 2\sqrt{5} + 5 = 12 + 5 + (6\sqrt{5} + 2\sqrt{5}) = 17 + 8\sqrt{5} \]

### Step 4: Combine the Results
Now we can combine the results from the numerator and denominator:

\[ \frac{17 + 8\sqrt{5}}{-1} = - (17 + 8\sqrt{5}) = -17 - 8\sqrt{5} \]

### Final Answer
Thus, the expression \((6 + \sqrt{5})/(2 - \sqrt{5})\) simplifies to:

\[ - (17 + 8\sqrt{5}) \]