When `(2x - 3)^(2)` is written in the form `ax^(2) + bx + c` where a, b and c are integers, a + b + c = ?

Options

A. `-17`
B. `-5`
C. `1`
D. `-1`

Answer

To solve the problem, we need to expand the expression \((2x - 3)^2\) and then identify the coefficients \(a\), \(b\), and \(c\) in the standard quadratic form \(ax^2 + bx + c\). Finally, we will calculate \(a + b + c\).

### Step-by-Step Solution:

1. **Start with the expression**:
\[ (2x - 3)^2 \]

2. **Use the formula for the square of a binomial**:
The formula for \((a - b)^2\) is \(a^2 - 2ab + b^2\). Here, \(a = 2x\) and \(b = 3\).

3. **Apply the formula**:
\[ (2x - 3)^2 = (2x)^2 - 2 \cdot (2x) \cdot 3 + 3^2 \]

4. **Calculate each term**:
- \((2x)^2 = 4x^2\)
- \(-2 \cdot (2x) \cdot 3 = -12x\)
- \(3^2 = 9\)

5. **Combine the terms**:
\[ (2x - 3)^2 = 4x^2 - 12x + 9 \]

6. **Identify coefficients**:
From the expression \(4x^2 - 12x + 9\), we can identify:
- \(a = 4\)
- \(b = -12\)
- \(c = 9\)

7. **Calculate \(a + b + c\)**:
\[ a + b + c = 4 + (-12) + 9 \]
\[ = 4 - 12 + 9 \]
\[ = -8 + 9 \]
\[ = 1 \]

### Final Answer:
Thus, \(a + b + c = 1\).