If a+b+c=p and ab+bc+ac=q, find `a^(2)+b^(2)+c^(2)`.

Answer

To solve the problem, we need to find \( a^2 + b^2 + c^2 \) given that \( a + b + c = p \) and \( ab + bc + ac = q \).

### Step-by-Step Solution:

1. **Use the identity for the square of a sum**:
We start with the identity:
\[ (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + ac + bc) \]
This identity relates the square of the sum of the variables to the sum of their squares and the products of the variables.

2. **Substitute known values**:
From the problem, we know:
\[ a + b + c = p \quad \text{and} \quad ab + ac + bc = q \]
Therefore, we can substitute these into the identity:
\[ p^2 = a^2 + b^2 + c^2 + 2q \]

3. **Rearrange the equation to solve for \( a^2 + b^2 + c^2 \)**:
We want to isolate \( a^2 + b^2 + c^2 \). Rearranging the equation gives us:
\[ a^2 + b^2 + c^2 = p^2 - 2q \]

4. **Final expression**:
Thus, the final expression for \( a^2 + b^2 + c^2 \) is:
\[ a^2 + b^2 + c^2 = p^2 - 2q \]

### Summary:
The result we derived is:
\[ a^2 + b^2 + c^2 = p^2 - 2q \]