# SOLUTION: The sum of the squares of two consecutive odd positive integers is 202. Find the integers.

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Question 255374: The sum of the squares of two consecutive odd positive integers is 202. Find the integers.
Found 3 solutions by drk, richwmiller, Alan3354:
You can put this solution on YOUR website!
let x = positive odd integer and x + 2 be the next positive odd integer.
we get
x^2 + (x+2)^2 = 202
foiling the left and combining like terms, we get
2x^2 + 4x + 4 = 202
and then
2x^2 + 4x - 198 = 0
divide by 2 to get
x^2 + 2x - 99 = 0
factor to get
(x+11)(x-9) = 0
solving for x, we get
x = -11 and x = 9
we get (-11,-9) OR (9,11)

You can put this solution on YOUR website!
n^2+(n+2)^2=202
2n^2+n+4=202
(2n+11)(n-9)=0
Be sure to follow how to factor.
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)
In order to factor , first multiply the leading coefficient 2 and the last term -198 to get -396. Now we need to ask ourselves: What two numbers multiply to -396 and add to 4? Lets find out by listing all of the possible factors of -396

Factors:

1,2,3,4,6,9,11,12,18,22,33,36,44,66,99,132,198,396,

-1,-2,-3,-4,-6,-9,-11,-12,-18,-22,-33,-36,-44,-66,-99,-132,-198,-396, List the negative factors as well. This will allow us to find all possible combinations

These factors pair up to multiply to -396.

(-1)*(396)=-396

(-2)*(198)=-396

(-3)*(132)=-396

(-4)*(99)=-396

(-6)*(66)=-396

(-9)*(44)=-396

(-11)*(36)=-396

(-12)*(33)=-396

(-18)*(22)=-396

Now which of these pairs add to 4? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 4

||||||||||||||||||
 First Number | Second Number | Sum 1 | -396 | 1+(-396)=-395 2 | -198 | 2+(-198)=-196 3 | -132 | 3+(-132)=-129 4 | -99 | 4+(-99)=-95 6 | -66 | 6+(-66)=-60 9 | -44 | 9+(-44)=-35 11 | -36 | 11+(-36)=-25 12 | -33 | 12+(-33)=-21 18 | -22 | 18+(-22)=-4 -1 | 396 | (-1)+396=395 -2 | 198 | (-2)+198=196 -3 | 132 | (-3)+132=129 -4 | 99 | (-4)+99=95 -6 | 66 | (-6)+66=60 -9 | 44 | (-9)+44=35 -11 | 36 | (-11)+36=25 -12 | 33 | (-12)+33=21 -18 | 22 | (-18)+22=4

We can see from the table that -18 and 22 add to 4. So the two numbers that multiply to -396 and add to 4 are: -18 and 22

breaks down to this (just replace with the two numbers that multiply to -396 and add to 4, which are: -18 and 22)

Replace with

Group the first two terms together and the last two terms together like this:

Factor a 2x out of the first group and factor a 22 out of the second group.

Now since we have a common term we can combine the two terms.

Combine like terms.
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