If `x=2+sqrt(2)` and `y=2-sqrt(2)`, then find the value of `(x^(2)+y^(2))`.

Options

A. 12
B. 14
C. 6
D. 18

Answer

To find the value of \( x^2 + y^2 \) given \( x = 2 + \sqrt{2} \) and \( y = 2 - \sqrt{2} \), we can follow these steps:

### Step 1: Calculate \( x^2 \)
Using the formula for squaring a binomial, we have:
\[ x^2 = (2 + \sqrt{2})^2 = 2^2 + 2 \cdot 2 \cdot \sqrt{2} + (\sqrt{2})^2 \]
Calculating each term:
- \( 2^2 = 4 \)
- \( 2 \cdot 2 \cdot \sqrt{2} = 4\sqrt{2} \)
- \( (\sqrt{2})^2 = 2 \)

So, combining these:
\[ x^2 = 4 + 4\sqrt{2} + 2 = 6 + 4\sqrt{2} \]

### Step 2: Calculate \( y^2 \)
Similarly, we calculate \( y^2 \):
\[ y^2 = (2 - \sqrt{2})^2 = 2^2 - 2 \cdot 2 \cdot \sqrt{2} + (\sqrt{2})^2 \]
Calculating each term:
- \( 2^2 = 4 \)
- \( -2 \cdot 2 \cdot \sqrt{2} = -4\sqrt{2} \)
- \( (\sqrt{2})^2 = 2 \)

So, combining these:
\[ y^2 = 4 - 4\sqrt{2} + 2 = 6 - 4\sqrt{2} \]

### Step 3: Add \( x^2 \) and \( y^2 \)
Now we can find \( x^2 + y^2 \):
\[ x^2 + y^2 = (6 + 4\sqrt{2}) + (6 - 4\sqrt{2}) \]
Combining like terms:
\[ x^2 + y^2 = 6 + 6 + 4\sqrt{2} - 4\sqrt{2} = 12 \]

### Final Answer
Thus, the value of \( x^2 + y^2 \) is \( \boxed{12} \).

---