` (4x - 7)^(2) = ? `
A. `4x^(2) - 56x +49`
B. `4x^(2) - 14 x +49`
C. `16 x^(2) +14 x +49`
D. ` 16 x^(2) - 56x +49`

Options

A. B
B. A
C. C
D. D

Answer

To solve the expression \((4x - 7)^{2}\), we will use the identity for the square of a binomial, which is given by:

\[ (a - b)^{2} = a^{2} - 2ab + b^{2} \]

In our case, we can identify \(a\) and \(b\) as follows:
- \(a = 4x\)
- \(b = 7\)

Now, we will apply the identity step by step.

### Step 1: Calculate \(a^{2}\)
\[ a^{2} = (4x)^{2} = 16x^{2} \]

### Step 2: Calculate \(-2ab\)
\[ -2ab = -2 \cdot (4x) \cdot 7 = -56x \]

### Step 3: Calculate \(b^{2}\)
\[ b^{2} = 7^{2} = 49 \]

### Step 4: Combine all the parts
Now we combine all the calculated parts:
\[ (4x - 7)^{2} = a^{2} - 2ab + b^{2} = 16x^{2} - 56x + 49 \]

Thus, the expanded form of \((4x - 7)^{2}\) is:
\[ 16x^{2} - 56x + 49 \]

### Final Answer
The correct answer is option D: \(16x^{2} - 56x + 49\).

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