The value of `(sqrt5 + 2)/(sqrt5 - 2)` is :

Options

A. `4 - 9sqrt5`
B. `9 + 4sqrt5`
C. `9 - 4sqrt5`
D. `7 + 4sqrt5`

Answer

To find the value of \(\frac{\sqrt{5} + 2}{\sqrt{5} - 2}\), we can rationalize the expression by multiplying the numerator and the denominator by the conjugate of the denominator. Here are the steps to solve the problem:

### Step 1: Write the expression
We start with the expression:
\[ \frac{\sqrt{5} + 2}{\sqrt{5} - 2} \]

### Step 2: Multiply by the conjugate
To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator, which is \(\sqrt{5} + 2\):
\[ \frac{(\sqrt{5} + 2)(\sqrt{5} + 2)}{(\sqrt{5} - 2)(\sqrt{5} + 2)} \]

### Step 3: Expand the numerator
Using the identity \((a + b)^2 = a^2 + 2ab + b^2\), we expand the numerator:
\[ (\sqrt{5} + 2)^2 = (\sqrt{5})^2 + 2 \cdot \sqrt{5} \cdot 2 + 2^2 = 5 + 4\sqrt{5} + 4 = 9 + 4\sqrt{5} \]

### Step 4: Expand the denominator
Using the identity \(a^2 - b^2 = (a - b)(a + b)\), we expand the denominator:
\[ (\sqrt{5})^2 - 2^2 = 5 - 4 = 1 \]

### Step 5: Combine the results
Now we can substitute back into our expression:
\[ \frac{9 + 4\sqrt{5}}{1} = 9 + 4\sqrt{5} \]

### Final Answer
Thus, the value of \(\frac{\sqrt{5} + 2}{\sqrt{5} - 2}\) is:
\[ \boxed{9 + 4\sqrt{5}} \]