If `x=(sqrt(5)-2)/(sqrt(5)+2) and y=(sqrt(5)+2)/(sqrt(5)-2):` find
(i) `x^(2)" (ii) "y^(2)`
(iii) xy `" (iv) "x^(2)+y^(2)+xy`

Answer

To solve the problem step by step, we will find \( x^2 \), \( y^2 \), \( xy \), and \( x^2 + y^2 + xy \) given:

\[ x = \frac{\sqrt{5} - 2}{\sqrt{5} + 2} \quad \text{and} \quad y = \frac{\sqrt{5} + 2}{\sqrt{5} - 2} \]

### Step 1: Calculate \( x^2 \)

We start with:

\[ x^2 = \left(\frac{\sqrt{5} - 2}{\sqrt{5} + 2}\right)^2 \]

Using the identity \( (a - b)^2 = a^2 - 2ab + b^2 \) and \( (a + b)^2 = a^2 + 2ab + b^2 \):

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

Simplifying:

\[ x^2 = \frac{9 - 4\sqrt{5}}{9 + 4\sqrt{5}} \]

### Step 2: Calculate \( y^2 \)

Now, we calculate \( y^2 \):

\[ y^2 = \left(\frac{\sqrt{5} + 2}{\sqrt{5} - 2}\right)^2 \]

Using the same identities:

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

Simplifying:

\[ y^2 = \frac{9 + 4\sqrt{5}}{9 - 4\sqrt{5}} \]

### Step 3: Calculate \( xy \)

Next, we find \( xy \):

\[ xy = \left(\frac{\sqrt{5} - 2}{\sqrt{5} + 2}\right) \left(\frac{\sqrt{5} + 2}{\sqrt{5} - 2}\right) \]

Using the identity \( (a - b)(a + b) = a^2 - b^2 \):

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

### Step 4: Calculate \( x^2 + y^2 + xy \)

Now we need to calculate \( x^2 + y^2 + xy \):

\[ x^2 + y^2 + xy = \frac{9 - 4\sqrt{5}}{9 + 4\sqrt{5}} + \frac{9 + 4\sqrt{5}}{9 - 4\sqrt{5}} + 1 \]

To combine these fractions, we need a common denominator:

\[ x^2 + y^2 + xy = \frac{(9 - 4\sqrt{5})^2 + (9 + 4\sqrt{5})^2 + (9 + 4\sqrt{5})(9 - 4\sqrt{5})}{(9 + 4\sqrt{5})(9 - 4\sqrt{5})} \]

Calculating the numerator:

1. \( (9 - 4\sqrt{5})^2 = 81 - 72\sqrt{5} + 80 = 161 - 72\sqrt{5} \)
2. \( (9 + 4\sqrt{5})^2 = 81 + 72\sqrt{5} + 80 = 161 + 72\sqrt{5} \)
3. \( (9 + 4\sqrt{5})(9 - 4\sqrt{5}) = 81 - 80 = 1 \)

Combining these:

\[ x^2 + y^2 + xy = \frac{(161 - 72\sqrt{5}) + (161 + 72\sqrt{5}) + 1}{(9 + 4\sqrt{5})(9 - 4\sqrt{5})} \]

This simplifies to:

\[ x^2 + y^2 + xy = \frac{323}{1} = 323 \]

### Final Answers:

1. \( x^2 = \frac{9 - 4\sqrt{5}}{9 + 4\sqrt{5}} \)
2. \( y^2 = \frac{9 + 4\sqrt{5}}{9 - 4\sqrt{5}} \)
3. \( xy = 1 \)
4. \( x^2 + y^2 + xy = 323 \)