SOLUTION: Find two consecutive positive odd numbers which are such that the square of their sum exceeds the sum of their square by 126.

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Question 1116316: Find two consecutive positive odd numbers which are such that the square of their sum exceeds the sum of their square by 126.
Answer by ikleyn(52882) About Me  (Show Source):
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Let x be the smallest positive odd number.

Then the second number is (x+2),  and the condition says


(2x+2)^2 = x^2 + (x+2)^2 + 126.


4x^2 + 8x + 4 = x^2 + x^2 + 2x + 4 + 126


2x^2 + 4x - 126 = 0


x^2 + 2x - 63 = 0


(x-7)*(x+9) = 0


The roots are  x= 7  and  x= -9.


Of them, only positive x= 7 satisfies the condition.


Answer. The two consecutive positive integers satisfying the condition are  7  and  9.

Solved.