2023 · Q80
If 2s = a + b + c, then what is s² + (s − a)(s − b) + (s − b)(s − c) + (s − c)(s − a) equal to?
Elementary MathematicsAlgebra
- A(a + b + c)²
- Bab + bc + ca
- C2(ab + bc + ca)
- D3(ab + bc + ca)
Explanation
Expand each product: (s−a)(s−b) = s² − s(a+b) + ab, and similarly for the other two. Their sum is 3s² − 2s(a+b+c) + (ab+bc+ca). Since 2s = a+b+c, this becomes 3s² − 4s² + (ab+bc+ca) = (ab+bc+ca) − s². Adding the leading s² gives ab + bc + ca. The answer is (B).