The sum of two numbers is 9 and the sum of their squares is 41. Find the numbers.

Let the first number be x.

Since the sum of two numbers is 9, so other number is 9 - x.

Given, the sum of squares of numbers = 41

⇒ x² + (9 - x)² = 41

⇒ x² + x² + 81 - 18x = 41

⇒ 2x² - 18x + 81 - 41 = 0

⇒ 2x² - 18x + 40 = 0

⇒ 2(x² - 9x + 20) = 0

⇒ x² - 9x + 20 = 0

⇒ x² - 4x - 5x + 20 = 0

⇒ x(x - 4) - 5(x - 4) = 0

⇒ (x - 4)(x - 5) = 0

⇒ x - 4 = 0 or x - 5 = 0

⇒ x = 4 or x = 5

∴ x = 5, 9 - x = 4

∴ x = 4, 9 - x = 5

Hence, the numbers are 4 and 5.