The value of `ab+bc+ca` if `a^2+b^2+c^2=30` and `a+b +c=6` is

Options

A. 1
B. 8
C. 3
D. 2

Answer

To find the value of \( ab + bc + ca \) given that \( a^2 + b^2 + c^2 = 30 \) and \( a + b + c = 6 \), we can use the identity:

\[ (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca) \]

### Step-by-step Solution:

1. **Write down the given equations:**
- \( a + b + c = 6 \)
- \( a^2 + b^2 + c^2 = 30 \)

2. **Square the sum \( a + b + c \):**
\[ (a + b + c)^2 = 6^2 = 36 \]

3. **Substitute the values into the identity:**
\[ 36 = a^2 + b^2 + c^2 + 2(ab + bc + ca) \]
Substitute \( a^2 + b^2 + c^2 = 30 \):
\[ 36 = 30 + 2(ab + bc + ca) \]

4. **Rearrange the equation to isolate \( ab + bc + ca \):**
\[ 36 - 30 = 2(ab + bc + ca) \]
\[ 6 = 2(ab + bc + ca) \]

5. **Divide both sides by 2:**
\[ ab + bc + ca = \frac{6}{2} = 3 \]

### Final Answer:
Thus, the value of \( ab + bc + ca \) is \( 3 \).