If `a + b + c = 11, a^2 + b^2 + c^2 = 51`, what is the value of `ab + bc + ac` ?

Options

A. 24
B. 28
C. 32
D. 35

Answer

To find the value of \( ab + ac + bc \) given the equations \( a + b + c = 11 \) and \( a^2 + b^2 + c^2 = 51 \), we can follow these steps:

### Step 1: Use the identity for the square of a sum
We know that:
\[ (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + ac + bc) \]
Substituting the known values into this identity, we have:
\[ 11^2 = 51 + 2(ab + ac + bc) \]

### Step 2: Calculate \( 11^2 \)
Calculating \( 11^2 \):
\[ 11^2 = 121 \]
So, we can rewrite the equation as:
\[ 121 = 51 + 2(ab + ac + bc) \]

### Step 3: Rearrange the equation
Now, we can rearrange the equation to isolate \( 2(ab + ac + bc) \):
\[ 121 - 51 = 2(ab + ac + bc) \]
This simplifies to:
\[ 70 = 2(ab + ac + bc) \]

### Step 4: Solve for \( ab + ac + bc \)
Now, divide both sides by 2:
\[ ab + ac + bc = \frac{70}{2} = 35 \]

Thus, the value of \( ab + ac + bc \) is \( 35 \).

### Final Answer
The value of \( ab + ac + bc \) is \( 35 \).

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