If `a^(2) + b^(2) + c^(2) = 50 and ab +bc + ca= 47`, find `a + b+ c`

Answer

To solve the problem, we need to find the value of \( a + b + c \) given the equations:

1. \( a^2 + b^2 + c^2 = 50 \)
2. \( ab + bc + ca = 47 \)

We can use the identity for the square of a sum:

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

### Step 1: Substitute the known values into the identity

We know:

- \( a^2 + b^2 + c^2 = 50 \)
- \( ab + bc + ca = 47 \)

Substituting these values into the identity gives us:

\[ (a + b + c)^2 = 50 + 2(47) \]

### Step 2: Calculate \( 2(ab + bc + ca) \)

Now calculate \( 2(47) \):

\[ 2(47) = 94 \]

### Step 3: Add the values together

Now substitute this back into the equation:

\[ (a + b + c)^2 = 50 + 94 \]

Calculating the right side:

\[ 50 + 94 = 144 \]

So we have:

\[ (a + b + c)^2 = 144 \]

### Step 4: Take the square root

To find \( a + b + c \), we take the square root of both sides:

\[ a + b + c = \sqrt{144} \]

Calculating the square root:

\[ \sqrt{144} = 12 \]

Thus, we have:

\[ a + b + c = 12 \quad \text{or} \quad a + b + c = -12 \]

Since \( a, b, c \) are typically considered as real numbers in this context, we take the positive value:

\[ a + b + c = 12 \]

### Final Answer

The value of \( a + b + c \) is \( 12 \).

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