What of the following is equivalent to `(a-5b)^2` ?

Options

A. 2a-10b
B. `a^2-25b^2`
C. `a^2-10ab+25b^2`
D. `a^2-12ab+25b^2`

Answer

To find the expression equivalent to \((a - 5b)^2\), we can use the algebraic identity for the square of a binomial. The identity states that:

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

### Step-by-Step Solution:

1. **Identify \(x\) and \(y\)**:
In our case, we can identify:
- \(x = a\)
- \(y = 5b\)

2. **Apply the formula**:
Using the formula \((x - y)^2 = x^2 - 2xy + y^2\), we substitute \(x\) and \(y\):
\[ (a - 5b)^2 = a^2 - 2(a)(5b) + (5b)^2 \]

3. **Calculate each term**:
- The first term is \(a^2\).
- The second term is \(-2(a)(5b) = -10ab\).
- The third term is \((5b)^2 = 25b^2\).

4. **Combine the terms**:
Putting it all together, we have:
\[ (a - 5b)^2 = a^2 - 10ab + 25b^2 \]

5. **Final expression**:
Thus, the expression equivalent to \((a - 5b)^2\) is:
\[ a^2 - 10ab + 25b^2 \]

### Conclusion:
The correct option from the given choices is:
- Option 3: \(a^2 - 10ab + 25b^2\)