If `a+b+ c =9 and a^(2) + b^(2) + c^(2) = 29`, find `ab + bc + ca`.

Answer

To solve the problem, we will use the identity related to the squares of sums. The identity states:

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

Given:
1. \( a + b + c = 9 \)
2. \( a^2 + b^2 + c^2 = 29 \)

We need to find \( ab + bc + ca \).

### Step 1: Calculate \( (a + b + c)^2 \)

Using the value of \( a + b + c \):

\[ (a + b + c)^2 = 9^2 = 81 \]

### Step 2: Substitute into the identity

Now, we substitute the known values into the identity:

\[ 81 = 29 + 2(ab + bc + ca) \]

### Step 3: Rearrange the equation

To isolate \( 2(ab + bc + ca) \), we subtract 29 from both sides:

\[ 81 - 29 = 2(ab + bc + ca) \]

Calculating the left side:

\[ 52 = 2(ab + bc + ca) \]

### Step 4: Solve for \( ab + bc + ca \)

Now, divide both sides by 2:

\[ ab + bc + ca = \frac{52}{2} = 26 \]

### Final Answer

Thus, the value of \( ab + bc + ca \) is \( 26 \).

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