If `a = 24, b = 26, c = 28`, then the value of `a^2 + b^2 + c^2 - ab - bc - ac` will be

Options

A. 0
B. 1
C. 8
D. 12

Answer

To solve the expression \( a^2 + b^2 + c^2 - ab - bc - ac \) given \( a = 24 \), \( b = 26 \), and \( c = 28 \), we will follow these steps:

### Step 1: Calculate \( a^2 \), \( b^2 \), and \( c^2 \)
- \( a^2 = 24^2 = 576 \)
- \( b^2 = 26^2 = 676 \)
- \( c^2 = 28^2 = 784 \)

### Step 2: Calculate \( ab \), \( bc \), and \( ac \)
- \( ab = 24 \times 26 = 624 \)
- \( bc = 26 \times 28 = 728 \)
- \( ac = 28 \times 24 = 672 \)

### Step 3: Substitute the values into the expression
Now we substitute these values into the expression:
\[ a^2 + b^2 + c^2 - ab - bc - ac = 576 + 676 + 784 - 624 - 728 - 672 \]

### Step 4: Calculate the sum of squares
First, calculate the sum of \( a^2 \), \( b^2 \), and \( c^2 \):
\[ 576 + 676 + 784 = 2036 \]

### Step 5: Calculate the sum of products
Now calculate the sum of \( ab \), \( bc \), and \( ac \):
\[ 624 + 728 + 672 = 2024 \]

### Step 6: Final calculation
Now we subtract the sum of products from the sum of squares:
\[ 2036 - 2024 = 12 \]

Thus, the value of \( a^2 + b^2 + c^2 - ab - bc - ac \) is \( 12 \).

### Final Answer:
\[ \text{The value is } 12. \]