If a + b + c = 6 and `a^2 + b^2 + c^2 = 26`, then what is ab+bc+ca equal to ??

Options

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

Answer

To solve the problem, we start with the given equations:

1. \( a + b + c = 6 \)
2. \( a^2 + b^2 + c^2 = 26 \)

We need to find the value of \( ab + bc + ca \).

### Step 1: Use the identity for the square of a sum
We can use the identity:
\[ (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca) \]

### Step 2: Substitute the known values
Substituting the known values into the identity:
\[ (6)^2 = 26 + 2(ab + bc + ca) \]
This simplifies to:
\[ 36 = 26 + 2(ab + bc + ca) \]

### Step 3: Rearrange the equation
Now, we can rearrange the equation to isolate \( ab + bc + ca \):
\[ 36 - 26 = 2(ab + bc + ca) \]
\[ 10 = 2(ab + bc + ca) \]

### Step 4: Solve for \( ab + bc + ca \)
Now, divide both sides by 2:
\[ ab + bc + ca = \frac{10}{2} = 5 \]

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

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