Question

The sum of the squares of two consecutive positive odd numbers is 514. Find the numbers.

Answer 1

Let the two consecutive positive odd numbers be x and (x + 2).

According to the given condition,

x² + (x + 2)² = 514 

⇒ x² + x² + 4x + 4 = 514 

⇒ 2x² + 4x – 510 = 0 

⇒ x² + 2x – 255 = 0 

⇒ x² + 17x – 15x – 255 = 0 

⇒ x(x + 17) – 15(x + 17) = 0 

⇒ (x + 17)(x – 15) = 0 

⇒ x + 17 = 0 or x – 15 = 0 

⇒ x = –17 or x = 15 

∴ x = 15   ...(x is a positive odd number) 

When x = 15, 

x + 2 = 15 + 2

= 17 

Hence, the required positive integers are 15 and 17.