Find two consecutive positive integers, sum of whose squares is 365.

Let the first number = x Therefore, second number = x + 1
according to the questions, x² + (x + 1)² = 365
⇒ x² + x² + 2x + 1 = 365 ⇒ 2x² + 2x – 364 = 0
⇒ x² + x – 182 = 0 ⇒ x² – 13x + 14x + 182 = 0
⇒ x(x – 13) + 14 (x – 13) = 0 ⇒ (x – 13) (x + 14) = 0
⇒ (x – 13) = 0 or (x + 14) = 0 ⇒ Either x = 13 or x = – 14
Hence, the two consecutive positive integers are 13 and 14.