SOLUTION: The sum of the square of three consecutive odd integers is 371. Find the odd integers

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Question 1120049: The sum of the square of three consecutive odd integers is 371. Find the odd integers



Found 3 solutions by math_helper, Alan3354, Shin123:
Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!
n%5E2+%2B+%28n%2B2%29%5E2+%2B+%28n%2B4%29%5E2+=+371+
+n%5E2+%2B+%28n%5E2%2B4n%2B4%29+%2B+%28n%5E2%2B8n%2B16%29+=+371+
+3n%5E2+%2B+12n+-+351+=+0+
+n%5E2+%2B+4n+-+117+=+0+
+%28n%2B13%29%28n-9%29+=+0+
n=9 and or n=-13
n=9: integers are 9,11,13
Check: +9%5E2+%2B+11%5E2+%2B+13%5E2+=+81%2B121%2B169+=+371+

and

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Tutor MathTherapy showed this method, which is a bit quicker, and perhaps less error-prone :
Let n = the middle integer
n-2 = smaller
n+2 = larger
+%28n-2%29%5E2+%2B+%28n%5E2%29+%2B+%28n%2B2%29%5E2+=+371+
++%28n%5E2-4n%2B4%29+%2B+n%5E2+%2B+%28n%5E2%2B4n%2B4%29+=+371+
++3n%5E2+-+363+=+0+
++n%5E2+-+121+=+0+
+%28n-11%29%28n%2B11%29+=+0+
n = 11 and/or n=-11 which gives the same solutions (9,11,13) and (-13,-11,-9).
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Very good method tutor Alan3354! Thanks.


Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
The sum of the square of three consecutive odd integers is 371. Find the odd integers
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371/3 = 123.666....
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sqrt(123) =~ 11, the middle number
---> 9, 11 & 13

Answer by Shin123(626) About Me  (Show Source):
You can put this solution on YOUR website!
%28n-4%29%5E2%2B%28n-2%29%5E2%2Bn%5E2=371
%28n%5E2-8n%2B16%29%2B%28n%5E2-4n%2B4%29%2Bn%5E2=371
3n%5E2-12n%2B20-371=0
3n%5E2-12n-351=0
n%5E2-4n-117=0
%28n-13%29%28n%2B9%29=0 n=13 or -9. The numbers are 9,11,13 or -13,-11,-9. Check:
9%5E2%2B11%5E2%2B13%5E2=371
81%2B121%2B169=371
371=371