If (a + b) = 12 and ab = 14, then `(a^(2) + b^(2)) = ?`

Options

A. 172
B. 116
C. 162
D. 126

Answer

To solve the problem, we need to find the value of \( a^2 + b^2 \) given that \( a + b = 12 \) and \( ab = 14 \). We can use the identity that relates these quantities.

### Step-by-Step Solution:

1. **Use the Identity**:
We know the identity:
\[ a^2 + b^2 = (a + b)^2 - 2ab \]

2. **Substitute the Given Values**:
We have \( a + b = 12 \) and \( ab = 14 \). Substitute these values into the identity:
\[ a^2 + b^2 = (12)^2 - 2 \times 14 \]

3. **Calculate \( (a + b)^2 \)**:
Calculate \( (12)^2 \):
\[ (12)^2 = 144 \]

4. **Calculate \( 2ab \)**:
Calculate \( 2 \times 14 \):
\[ 2 \times 14 = 28 \]

5. **Substitute and Simplify**:
Now substitute these values back into the equation:
\[ a^2 + b^2 = 144 - 28 \]

6. **Final Calculation**:
Perform the subtraction:
\[ a^2 + b^2 = 116 \]

Thus, the value of \( a^2 + b^2 \) is \( 116 \).

### Final Answer:
\[ a^2 + b^2 = 116 \]

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