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Question
Find two consecutive positive integers, sum of whose squares is 365.
Solution
Product of two consecutive positive integers = 306
Let first positive integer = x
Second positive integer = x + 1
Sum of squares of both consecutive numbers = 365
(x)2 + (x + 1)2 = 365
x2 + x2 + 2x + 1 = 365
2x2 + 2x + 1 = 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 + 14)(x - 13) = 0
x + 14 = 0 and x - 13 = 0
x = -14 and x = 13
Since
First positive integer = x = 13
Second positive integer = x + 1 = 13 + 1 = 14
Thus, the required consecutive positive integers are 13 and 14.
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