SOLUTION: The sum of the squares of four consecutive odd integers is 36. What is the difference between the largest and smallest of the integers?

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Question 972223: The sum of the squares of four consecutive odd integers is 36. What is the difference between the largest and smallest of the integers?
Found 2 solutions by josgarithmetic, MathTherapy:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
n is any integer number.

%282n%2B1%29%5E2%2B%282n%2B3%29%5E2%2B%282n%2B5%29%5E2%2B%282n%2B7%29%5E2=36, and you are asked for what is %282n%2B7%29-%282n-1%29.



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To be sure,
4n%5E2%2B4n%2B1%2B4n%5E2%2B12n%2B9%2B4n%5E2%2B20n%2B25%2B4n%5E2%2B28n%2B49=36
16n%5E2%2B64n%2B84=36
16n%5E2%2B64n%2B48=0
n%5E2%2B4n%2B3=0
%28n%2B1%29%28n%2B3%29=0
n is either -1 or -3.
NOT both at the same time: Must be one or the other.

Taking n=-1,
largest integer of the set:2(-1)+7=5.
smallest integer of the set: 2(-1)+1=-1.
DIFFERENCE REQUESTED: highlight%286%29.
Taking n=-3,
DIFFERENCE REQUESTED: %282%28-3%29%2B7%29-%282%28-3%29%2B1%29=1-%28-5%29=highlight%286%29.

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!
The sum of the squares of four consecutive odd integers is 36. What is the difference between the largest and smallest of the integers?
Let the least of the numbers be S

Then other 3 are: S + 2, S + 4, and S + 6

We then get: S%5E2+%2B+%28S+%2B+2%29%5E2+%2B+%28S+%2B+4%29%5E2+%2B+%28S+%2B+6%29%5E2=+36

S%5E2+%2B+S%5E2+%2B+4S+%2B+4+%2B+S%5E2+%2B+8S+%2B+16+%2B+S%5E2+%2B+12S+%2B+36+=+36+

4S%5E2+%2B+24S+%2B+56+=+36

4S%5E2+%2B+24S+%2B+56+-+36+=+0

4S%5E2+%2B+24S+%2B+20+=+0

4%28S%5E2+%2B+6S+%2B+5%29+=+4%280%29

S%5E2+%2B+6S+%2B+5+=+0

(S + 5)(S + 1) = 0

S, or smallest integer = - 5              OR                S = - 1



If smallest integer is: - 5, then 4 integers are: highlight%28system%28-+5%2C-+3%2C+-+1%2C+1%29%29

Difference between largest and smallest integer: 1 - - 5, or 1 + 5, or highlight_green%286%29



If smallest integer is: - 1, then 4 integers are: highlight%28system%28-+1%2C+1%2C+3%2C+5%29%29

Difference between largest and smallest integer: 5 - - 1, or 5 + 1, or highlight_green%286%29