If x `=(sqrt(5)-2)/(sqrt(5)+2)` and `y = (sqrt(5)+2)/(sqrt(5)-2)` : find :
`x^(2)`

Answer

To find \( x^2 \) given \( x = \frac{\sqrt{5} - 2}{\sqrt{5} + 2} \), we will follow these steps:

### Step 1: Write the expression for \( x^2 \)

We start with the expression for \( x \):
\[ x = \frac{\sqrt{5} - 2}{\sqrt{5} + 2} \]
To find \( x^2 \), we square this expression:
\[ x^2 = \left( \frac{\sqrt{5} - 2}{\sqrt{5} + 2} \right)^2 \]

### Step 2: Square the numerator and the denominator

Squaring the fraction gives us:
\[ x^2 = \frac{(\sqrt{5} - 2)^2}{(\sqrt{5} + 2)^2} \]

Now we calculate the squares of the numerator and denominator separately.

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

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

### Step 3: Combine the results

Now we can substitute the results back into our expression for \( x^2 \):
\[ x^2 = \frac{9 - 4\sqrt{5}}{9 + 4\sqrt{5}} \]

### Step 4: Rationalize the denominator

To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator:
\[ x^2 = \frac{(9 - 4\sqrt{5})(9 - 4\sqrt{5})}{(9 + 4\sqrt{5})(9 - 4\sqrt{5})} \]

**Numerator:**
\[ (9 - 4\sqrt{5})^2 = 9^2 - 2 \cdot 9 \cdot 4\sqrt{5} + (4\sqrt{5})^2 = 81 - 72\sqrt{5} + 80 = 161 - 72\sqrt{5} \]

**Denominator:**
Using the difference of squares:
\[ (9 + 4\sqrt{5})(9 - 4\sqrt{5}) = 9^2 - (4\sqrt{5})^2 = 81 - 80 = 1 \]

### Step 5: Final result

Thus, we have:
\[ x^2 = 161 - 72\sqrt{5} \]