If `a+b+c=6, a^(2)+b^(2)+c^(2)=26`, then `ab+bc+ca` is equal to

Options

A. 0
B. 2
C. 4
D. 5

Answer

To solve the problem, we need to find the value of \( ab + bc + ca \) given that \( a + b + c = 6 \) and \( a^2 + b^2 + c^2 = 26 \).

### Step-by-Step Solution:

1. **Use the Identity**:
We start with the algebraic identity:
\[ (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca) \]

2. **Substitute the Known Values**:
We know that \( a + b + c = 6 \) and \( a^2 + b^2 + c^2 = 26 \). Substituting these values into the identity:
\[ 6^2 = 26 + 2(ab + bc + ca) \]

3. **Calculate \( 6^2 \)**:
Calculate \( 6^2 \):
\[ 36 = 26 + 2(ab + bc + ca) \]

4. **Rearranging the Equation**:
Now, we rearrange the equation to isolate \( 2(ab + bc + ca) \):
\[ 36 - 26 = 2(ab + bc + ca) \]
\[ 10 = 2(ab + bc + ca) \]

5. **Divide by 2**:
To find \( ab + bc + ca \), divide both sides by 2:
\[ ab + bc + ca = \frac{10}{2} = 5 \]

### Final Answer:
Thus, the value of \( ab + bc + ca \) is \( 5 \).