The value of `(a + b) ^(2) + (a-b) ^(2)` is

Options

A. `2a + 2b`
B. `2a - 2b`
C. `2a ^(2) + 2b ^(2)`
D. `2a ^(2) - 2b ^(2)`

Answer

To find the value of \((a + b)^2 + (a - b)^2\), we will expand each term using the algebraic identities for squares of binomials.

### Step-by-Step Solution:

1. **Expand \((a + b)^2\)**:
\[ (a + b)^2 = a^2 + 2ab + b^2 \]

2. **Expand \((a - b)^2\)**:
\[ (a - b)^2 = a^2 - 2ab + b^2 \]

3. **Add the two expansions together**:
\[ (a + b)^2 + (a - b)^2 = (a^2 + 2ab + b^2) + (a^2 - 2ab + b^2) \]

4. **Combine like terms**:
\[ = a^2 + 2ab + b^2 + a^2 - 2ab + b^2 \]
\[ = a^2 + a^2 + b^2 + b^2 + 2ab - 2ab \]
\[ = 2a^2 + 2b^2 \]

5. **Final Result**:
\[ (a + b)^2 + (a - b)^2 = 2a^2 + 2b^2 \]