What is the value of `(203 + 107)^(2) - (203-107)^(2)?`

Options

A. 85886
B. 86884
C. 43442
D. 87884

Answer

To solve the expression \( (203 + 107)^{2} - (203 - 107)^{2} \), we can use the algebraic identity for the difference of squares.

### Step-by-Step Solution:

1. **Identify the expression**:
We have \( (203 + 107)^{2} - (203 - 107)^{2} \).

2. **Let \( A = 203 \) and \( B = 107 \)**:
This gives us \( (A + B)^{2} - (A - B)^{2} \).

3. **Use the difference of squares formula**:
The difference of squares states that \( x^{2} - y^{2} = (x - y)(x + y) \).
Here, let \( x = A + B \) and \( y = A - B \).

4. **Calculate \( x - y \) and \( x + y \)**:
- \( x - y = (A + B) - (A - B) = A + B - A + B = 2B \).
- \( x + y = (A + B) + (A - B) = A + B + A - B = 2A \).

5. **Substitute back into the difference of squares**:
Therefore, we have:
\[ (A + B)^{2} - (A - B)^{2} = (2B)(2A) = 4AB. \]

6. **Substitute the values of \( A \) and \( B \)**:
Now, substitute \( A = 203 \) and \( B = 107 \):
\[ 4AB = 4 \times 203 \times 107. \]

7. **Calculate \( 4 \times 203 \times 107 \)**:
- First, calculate \( 203 \times 107 \):
\[ 203 \times 107 = 21721. \]
- Now multiply by 4:
\[ 4 \times 21721 = 86884. \]

8. **Final Result**:
Thus, the value of \( (203 + 107)^{2} - (203 - 107)^{2} \) is \( 86884 \).

### Final Answer:
The value is \( 86884 \).

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