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

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

Concept/Approach
Use 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/Alternative
Algebraically 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 Answer
20