If `a^(2)+b^(2)+c^(2)=50andab+bc+ca=47`, find the value of `a+b+c`.

Answer

To find the value of \( a + b + c \) given the equations \( a^2 + b^2 + c^2 = 50 \) and \( ab + bc + ca = 47 \), we can use the algebraic identity:

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

### Step-by-Step Solution:

1. **Write down the identity:**
\[ (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca) \]

2. **Substitute the known values:**
We know that \( a^2 + b^2 + c^2 = 50 \) and \( ab + bc + ca = 47 \). Substitute these values into the identity:
\[ (a + b + c)^2 = 50 + 2 \times 47 \]

3. **Calculate \( 2 \times 47 \):**
\[ 2 \times 47 = 94 \]

4. **Add the values:**
\[ (a + b + c)^2 = 50 + 94 = 144 \]

5. **Take the square root:**
To find \( a + b + c \), take the square root of both sides:
\[ a + b + c = \sqrt{144} \]

6. **Calculate the square root:**
\[ \sqrt{144} = 12 \]

Thus, the value of \( a + b + c \) is \( 12 \).

### Final Answer:
\[ a + b + c = 12 \]