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

Answer

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

### Step-by-Step Solution:

1. **Start with the identity for the square of a sum**:
\[ (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + ac + bc) \]

2. **Substitute the known values**:
Since \( a + b + c = p \), we can substitute \( p \) into the equation:
\[ p^2 = a^2 + b^2 + c^2 + 2(ab + ac + bc) \]

3. **Substitute \( ab + ac + bc \) with \( q \)**:
We know that \( ab + ac + bc = q \). Therefore, we can rewrite the equation as:
\[ p^2 = a^2 + b^2 + c^2 + 2q \]

4. **Rearrange to solve for \( a^2 + b^2 + c^2 \)**:
To isolate \( a^2 + b^2 + c^2 \), we subtract \( 2q \) from both sides:
\[ a^2 + b^2 + c^2 = p^2 - 2q \]

5. **Final expression**:
Thus, we find that:
\[ a^2 + b^2 + c^2 = p^2 - 2q \]

### Final Answer:
\[ a^2 + b^2 + c^2 = p^2 - 2q \]

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