Question #4919

the sum of the squares of 2 consecutive natural numbers is 313 find the number

Expert's answer

Let first positive integer is x. second is (x+1).

x^2 +(x+1)^2 = 313

x^2 + x^2 + 2x + 1 = 313

2x^2 + 2x - 312 = 0

x^2 + x - 156 = 0

x^2 + 13x - 12x - 156 = 0

x(x+13) - 12(x+13) = 0

(x+13)(x-12) = 0

x = -13 or x = 12

x =-13 rejected,

x = 12, and x+1 = 12+1 = 13

So, two consecutive positive integers are 12 and 13.

x^2 +(x+1)^2 = 313

x^2 + x^2 + 2x + 1 = 313

2x^2 + 2x - 312 = 0

x^2 + x - 156 = 0

x^2 + 13x - 12x - 156 = 0

x(x+13) - 12(x+13) = 0

(x+13)(x-12) = 0

x = -13 or x = 12

x =-13 rejected,

x = 12, and x+1 = 12+1 = 13

So, two consecutive positive integers are 12 and 13.

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