The sum of the squares of three numbers is 86, and the sum of their products taken two at a time is 75. What is the sum of the three numbers?

Question: The sum of the squares of three numbers is 86, and the sum of their products taken two at a time is 75. What is the sum of the three numbers?

Solution:
Let the numbers be a, b, c.
a² + b² + c² = 86,
ab + bc + ca = 75

We know,
(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

Substitute the given values:
(a + b + c)² = 86 + 2 × 75
= 86 + 150
= 236
∴ a + b + c = √236