The value of `(sqrt(5) - 2)/(sqrt(5) + 2)` is :

Options

A. `4 - 9sqrt(5)`
B. `9 + 4sqrt(5)`
C. `9 - 4sqrt(5)`
D. `7 + 4sqrt(5)`

Answer

To find the value of \((\sqrt{5} - 2)/(\sqrt{5} + 2)\), we will rationalize the denominator. Here are the steps:

### Step 1: Multiply by the Conjugate
We will multiply the numerator and the denominator by the conjugate of the denominator, which is \((\sqrt{5} - 2)\).

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

### Step 2: Simplify the Denominator
Using the difference of squares formula \(a^2 - b^2\), where \(a = \sqrt{5}\) and \(b = 2\):

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

### Step 3: Expand the Numerator
Now, we will expand the numerator \((\sqrt{5} - 2)^2\):

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

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

\[ \frac{9 - 4\sqrt{5}}{1} = 9 - 4\sqrt{5} \]

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

\[ \boxed{9 - 4\sqrt{5}} \]