If `(3+2sqrt(5))/(3-2sqrt(5))=(p+qsqrt(5))`, then find the value of `11(p+q)`.

Options

A. `31`
B. `-41`
C. `-31`
D. `-40`

Answer

To solve the equation \(\frac{3 + 2\sqrt{5}}{3 - 2\sqrt{5}} = p + q\sqrt{5}\), we will follow these steps:

### Step 1: Rationalize the denominator
To eliminate the square root in the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator, which is \(3 + 2\sqrt{5}\):

\[ \frac{(3 + 2\sqrt{5})(3 + 2\sqrt{5})}{(3 - 2\sqrt{5})(3 + 2\sqrt{5})} \]

### Step 2: Calculate the denominator
Using the difference of squares formula \(a^2 - b^2\):

\[ (3 - 2\sqrt{5})(3 + 2\sqrt{5}) = 3^2 - (2\sqrt{5})^2 = 9 - 4 \cdot 5 = 9 - 20 = -11 \]

### Step 3: Calculate the numerator
Now we calculate the numerator:

\[ (3 + 2\sqrt{5})(3 + 2\sqrt{5}) = 3^2 + 2 \cdot 3 \cdot 2\sqrt{5} + (2\sqrt{5})^2 = 9 + 12\sqrt{5} + 4 \cdot 5 = 9 + 12\sqrt{5} + 20 = 29 + 12\sqrt{5} \]

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

\[ \frac{29 + 12\sqrt{5}}{-11} = -\frac{29}{11} - \frac{12}{11}\sqrt{5} \]

### Step 5: Identify \(p\) and \(q\)
From the expression \(-\frac{29}{11} - \frac{12}{11}\sqrt{5}\), we can identify:

- \(p = -\frac{29}{11}\)
- \(q = -\frac{12}{11}\)

### Step 6: Calculate \(11(p + q)\)
Now we need to find \(11(p + q)\):

\[ p + q = -\frac{29}{11} - \frac{12}{11} = -\frac{29 + 12}{11} = -\frac{41}{11} \]

Now, multiplying by 11:

\[ 11(p + q) = 11 \cdot \left(-\frac{41}{11}\right) = -41 \]

### Final Answer
Thus, the value of \(11(p + q)\) is \(-41\).

---