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Mathematics

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

Quadratic Equations

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Answer

Let two consecutive positive integers be x and (x + 1).

According to question,

⇒ x2 + (x + 1)2 = 365

⇒ x2 + x2 + 1 + 2x = 365

⇒ 2x2 + 2x + 1 - 365 = 0

⇒ 2x2 + 2x - 364 = 0

⇒ 2(x2 + x - 182) = 0

⇒ x2 + x - 182 = 0

⇒ x2 + 14x - 13x - 182 = 0

⇒ x(x + 14) - 13(x + 14) = 0

⇒ (x - 13)(x + 14) = 0

⇒ x - 13 = 0 or x + 14 = 0

⇒ x = 13 or x = -14.

Since, positive integer cannot be negative.

∴ x = 13 and x + 1 = 14.

Hence, required positive integers are 13 and 14.

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