If `a^(2) + b^(2) + c^(2) = 35 and ab+ bc + ca= 23`, find `a + b+ c`

Answer

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

1. \( a^2 + b^2 + c^2 = 35 \)
2. \( ab + bc + ca = 23 \)

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

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

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

Using the values from the problem:

\[ (a + b + c)^2 = 35 + 2 \times 23 \]

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

Calculate \( 2 \times 23 \):

\[ 2 \times 23 = 46 \]

### Step 3: Add the values

Now, substitute this back into the equation:

\[ (a + b + c)^2 = 35 + 46 \]

### Step 4: Calculate the sum

Now, calculate \( 35 + 46 \):

\[ 35 + 46 = 81 \]

### Step 5: Take the square root

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

\[ a + b + c = \sqrt{81} \]

### Step 6: Final calculation

Calculate \( \sqrt{81} \):

\[ \sqrt{81} = 9 \]

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

\[ \boxed{9} \]