Factorize `a^(2) - (b-c)^(2)`.

Options

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

Answer

To factorize the expression \( a^2 - (b - c)^2 \), we can use the difference of squares formula, which states that:

\[ x^2 - y^2 = (x - y)(x + y) \]

In our case, we can identify \( x \) and \( y \) as follows:

- Let \( x = a \)
- Let \( y = (b - c) \)

Now we can apply the difference of squares formula:

1. **Identify the terms**:
\[ a^2 - (b - c)^2 \]

2. **Apply the difference of squares formula**:
\[ = (a - (b - c))(a + (b - c)) \]

3. **Simplify the expression**:
- For \( a - (b - c) \):
\[ a - (b - c) = a - b + c \]
- For \( a + (b - c) \):
\[ a + (b - c) = a + b - c \]

4. **Combine the simplified terms**:
\[ = (a - b + c)(a + b - c) \]

Thus, the factorized form of \( a^2 - (b - c)^2 \) is:

\[ (a - b + c)(a + b - c) \]