Question

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

Answer 1

Let the consecutive positive integers be x and x + 1.

Therefore, x² + (x + 1)² = 365

⇒ x² + x² + 1 + 2x = 365

⇒ 2x² + 2x - 364 = 0

⇒ x² + x - 182 = 0

⇒ x² + 14x - 13x - 182 = 0

⇒ x(x + 14) -13(x + 14) = 0

⇒ (x + 14)(x - 13) = 0

Either x + 14 = 0 or x - 13 = 0,

⇒ x = - 14 or x = 13

Since the integers are positive, x can only be 13.

∴ x + 1 = 13 + 1 = 14
Therefore, two consecutive positive integers will be 13 and 14.