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:
Answer by drk(1908)   (Show Source):
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)
Answer by richwmiller(9143)  (Show Source):
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
So the original quadratic
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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Answer:
So the quadratic factors to
Notice how foils back to our original problem . This verifies our answer. | |
Answer by Alan3354(30993)  (Show Source):
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