The sum of the squares of three numbers is 123, and the sum of their products taken two at a time is 119. What is the sum of the three numbers?

Question: The sum of the squares of three numbers is 123, and the sum of their products taken two at a time is 119. What is the sum of the three numbers?

Solution:
Let the numbers be a, b, c.
Given: a²+ b² + c² = 123, ab + bc + ca = 119
⇒ (a + b + c)² = a² + b² + c² + 2(ab + bc + ca) = 123 + 2 × 119 = 123 + 238 = 361

∴ a + b + c = √361 = 19