The value of `(4.669)^(2)+(2.331)^(2)+14(0.667)(2.331)` is (1-k) where k = ?

Options

A. -48
B. 0.647
C. 49
D. 0.768

Answer

To solve the equation \( (4.669)^2 + (2.331)^2 + 14(0.667)(2.331) = 1 - k \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the terms**:
We have three terms on the left side of the equation:
- \( (4.669)^2 \)
- \( (2.331)^2 \)
- \( 14(0.667)(2.331) \)

2. **Rewrite the third term**:
Notice that \( 14(0.667)(2.331) \) can be rewritten. Since \( 0.667 \) is approximately \( \frac{2.331}{7} \), we can express it as:
\[ 14(0.667)(2.331) = 14 \cdot \frac{2.331}{7} \cdot 2.331 = 4(2.331)^2 \]

3. **Combine the terms**:
Now we can rewrite the entire expression:
\[ (4.669)^2 + (2.331)^2 + 4(2.331)^2 = (4.669)^2 + 5(2.331)^2 \]

4. **Use the identity**:
Recognize that \( (4.669)^2 + (2.331)^2 + 2 \cdot 4.669 \cdot 2.331 \) resembles the square of a binomial:
\[ (a + b)^2 = a^2 + b^2 + 2ab \]
Here, if we let \( a = 4.669 \) and \( b = 2.331 \), we can express the left side as:
\[ (4.669 + 2.331)^2 = (7)^2 \]

5. **Substitute back into the equation**:
Now substitute back into the equation:
\[ 7^2 = 1 - k \]
This simplifies to:
\[ 49 = 1 - k \]

6. **Solve for \( k \)**:
Rearranging gives:
\[ k = 1 - 49 = -48 \]

### Final Answer:
Thus, the value of \( k \) is \( -48 \).