The sum of two numbers is 8 and the difference of their squares is 32. Find the numbers.

Let two numbers be x and y, such that x y. Given, Sum of two numbers = 8 ⇒ x + y = 8 ⇒ x = 8 - y ........(1) Given, Difference of squares = 32 ⇒ x 2 - y 2 = 32 ⇒ (8 - y) 2 - y 2 = 32 ⇒ 8 2 + y 2 - 2 × 8 × y - y 2 = 32 ⇒ 64 - 16y = 32 ⇒ 16y = 64 - 32 ⇒ 16y = 32 ⇒ y = = 2. Substituting value of y in equation (1), we get : ⇒ x = 8 - 2 = 6. Hence, numbers are 2 and 6.