If `a + b + c= 12 and a^(2) + b^(2) + c^(2) = 50`, find `ab + bc + ca`

Answer

To solve the problem, we will use the given equations and the formula for the square of a sum. Here are the steps:

1. **Write down the given equations:**
\[ a + b + c = 12 \quad \text{(1)} \]
\[ a^2 + b^2 + c^2 = 50 \quad \text{(2)} \]

2. **Use the formula for the square of a sum:**
The formula states that:
\[ (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca) \]

3. **Substitute the known values into the formula:**
From equation (1), we have:
\[ (12)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca) \]
This simplifies to:
\[ 144 = 50 + 2(ab + bc + ca) \]

4. **Rearrange the equation to isolate \(2(ab + bc + ca)\):**
\[ 144 - 50 = 2(ab + bc + ca) \]
\[ 94 = 2(ab + bc + ca) \]

5. **Divide both sides by 2 to find \(ab + bc + ca\):**
\[ ab + bc + ca = \frac{94}{2} = 47 \]

Thus, the value of \(ab + bc + ca\) is **47**.