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Two consecutive positive integers, the sum of whose square is 365, are:

Two consecutive positive integers, the sum of whose square is 365, are: Correct Answer 13, 14

Concept: 

Quadratic Equations: In order to factorize the quadratic expression Ax2 + Bx + C, we need to split the middle term B as a sum of two numbers whose product also equals the product AC.

Let's say that Ax2 + Bx + C = (x - α)(x - β) = 0.

Then the solution is, either x - α = 0 or x - β = 0 ⇒ x = α or x = β.

Calculation: 

Let's say the smaller number is x. Then the other number will be (x + 1).

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

⇒ x2 + x2 + 2x + 1 = 365

⇒ 2x2 + 2x - 364 = 0

⇒ x2 + x - 182 = 0

⇒ x2 + (14 - 13)x - 14 × 13 = 0

⇒ (x2 + 14x) - (13x + 14 × 13) = 0

⇒ x(x + 14) - 13(x + 14) = 0

⇒ (x + 14)(x - 13) = 0

⇒ x = -14 or x = 13

Since x is given to be a positive integer, the numbers are x = 13 and (x + 1) = 13 + 1 = 14.

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