The sum of the squares of three numbers is 138, and the sum of their pairwise products is 131. What is their sum?

Problem restatementLet the numbers be x, y, z. Given x² + y² + z² = 138 and xy + yz + zx = 131. Find S = x + y + z.

Concept/ApproachUse the identity S² = (x + y + z)² = x² + y² + z² + 2(xy + yz + zx).

Step-by-step calculation S² = 138 + 2×131 = 138 + 262 = 400 S = √400 = 20 (taking the principal value)

Verification/AlternativeAlgebraically S could be −20 as well, but unless specified otherwise, the conventional answer reported is 20.

Common pitfalls

• Forgetting the factor of 2 on the sum of pairwise products.

Final Answer20