Q. 44.3( 268 Votes )

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

Answer :

To Find: Consecutive integers sum of whose square is 365
Consecutive integers mean that the difference between the integers is of 1
Let the consecutive positive integers be x and x + 1.

Given that x2 + (x + 1)2 = 365


⇒ x2 + x2 + 1 + 2x = 365


⇒ 2x2 + 2x – 364 = 0


⇒  x2 + x – 182 = 0


Now to factorize the above quadratic equation, we need to chose numbers such that their product is 182 and difference is 1

⇒ 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, i.e., x = 14 or x = 13


Since the question ask for positive integers, x can only be 13.


x + 1 = 13 + 1 = 14


Therefore, two consecutive positive integers will be 13 and 14.

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