The sum of two natural numbers is 12 and the sum of their squares is 74. Find the numbers.

Let the numbers be x and y.

Given,

Sum of numbers = 12.

⇒ x + y = 12

⇒ x = 12 - y     ………(1)

Given,

Sum of their squares is 74.

⇒ x² + y² = 74     ………(2)

Substituting value of x from equation (1) in equation (2), we get :

⇒ (12 - y)² + y² = 74

⇒ (12)² + y² - 2 × 12 × y + y² = 74

⇒ y² + y² - 24y + 144 = 74

⇒ 2y² - 24y + 144 - 74 = 0

⇒ 2y² - 24y + 70 = 0

⇒ 2y² - 10y - 14y + 70 = 0

⇒ 2y(y - 5) - 14(y - 5) = 0

⇒ (2y - 14) (y - 5) = 0

⇒ (2y - 14) = 0 or (y - 5) = 0     [Using zero-product rule]

⇒ 2y = 14 or y = 5

⇒ y = $\dfrac{14}{2}$ or y = 5

⇒ y = 7 or y = 5

Substituting value of y in equation (1), we get :

Case 1: If y = 7,

x = 12 - 7 = 5.

Case 2: If y = 5,

x = 12 - 5 = 7.

Hence, the two natural numbers are 5 and 7.