If ` a=1 + sqrt3, b = 1 - sqrt3,` then what is value of `a ^(2) + b ^(2)` ?

Options

A. 4
B. 8
C. 0
D. 2

Answer

To solve the problem, we need to find the value of \( a^2 + b^2 \) given that \( a = 1 + \sqrt{3} \) and \( b = 1 - \sqrt{3} \).

### Step-by-Step Solution:

1. **Calculate \( a + b \)**:
\[ a + b = (1 + \sqrt{3}) + (1 - \sqrt{3}) \]
Simplifying this:
\[ a + b = 1 + \sqrt{3} + 1 - \sqrt{3} = 2 \]

2. **Calculate \( ab \)**:
\[ ab = (1 + \sqrt{3})(1 - \sqrt{3}) \]
Using the difference of squares formula:
\[ ab = 1^2 - (\sqrt{3})^2 = 1 - 3 = -2 \]

3. **Use the identity \( a^2 + b^2 = (a + b)^2 - 2ab \)**:
\[ a^2 + b^2 = (a + b)^2 - 2ab \]
Substitute the values we found:
\[ a^2 + b^2 = (2)^2 - 2(-2) \]

4. **Calculate \( (2)^2 \)**:
\[ (2)^2 = 4 \]

5. **Calculate \( -2(-2) \)**:
\[ -2(-2) = 4 \]

6. **Combine the results**:
\[ a^2 + b^2 = 4 + 4 = 8 \]

Thus, the value of \( a^2 + b^2 \) is \( 8 \).

### Final Answer:
\[ \boxed{8} \]