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

Answer

To solve the problem, we will follow these steps:

1. **Given Equations**: We have two equations:
- \( a^2 + b^2 + c^2 = 29 \) (Equation 1)
- \( a + b + c = 9 \) (Equation 2)

2. **Square Equation 2**: We will square the second equation:
\[ (a + b + c)^2 = 9^2 \]
This gives us:
\[ a^2 + b^2 + c^2 + 2(ab + bc + ca) = 81 \]

3. **Substitute Equation 1 into the Squared Equation**: We know from Equation 1 that \( a^2 + b^2 + c^2 = 29 \). We substitute this into the squared equation:
\[ 29 + 2(ab + bc + ca) = 81 \]

4. **Isolate \( ab + bc + ca \)**: Now, we will isolate \( ab + bc + ca \):
\[ 2(ab + bc + ca) = 81 - 29 \]
\[ 2(ab + bc + ca) = 52 \]

5. **Divide by 2**: Finally, we divide both sides by 2 to find \( ab + bc + ca \):
\[ ab + bc + ca = \frac{52}{2} = 26 \]

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