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(17219)  (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) |
 Start with the given expression.
 Factor out the GCF  .
Now let's try to factor the inner expression 
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Looking at the expression , we can see that the first coefficient is  , the second coefficient is , and the last term is  .
Now multiply the first coefficient by the last term to get  .
Now the question is: what two whole numbers multiply to (the previous product) and add to the second coefficient ?
To find these two numbers, we need to list all of the factors of (the previous product).
Factors of :
1,3,9,11,33,99
-1,-3,-9,-11,-33,-99
Note: list the negative of each factor. This will allow us to find all possible combinations.
These factors pair up and multiply to .
1*(-99) = -99
3*(-33) = -99
9*(-11) = -99
(-1)*(99) = -99
(-3)*(33) = -99
(-9)*(11) = -99
Now let's add up each pair of factors to see if one pair adds to the middle coefficient :
| First Number | Second Number | Sum | | 1 | -99 | 1+(-99)=-98 | | 3 | -33 | 3+(-33)=-30 | | 9 | -11 | 9+(-11)=-2 | | -1 | 99 | -1+99=98 | | -3 | 33 | -3+33=30 | | -9 | 11 | -9+11=2 |
From the table, we can see that the two numbers  and  add to (the middle coefficient).
So the two numbers and both multiply to and add to
Now replace the middle term  with  . Remember, and add to . So this shows us that  .
 Replace the second term with .
 Group the terms into two pairs.
 Factor out the GCF  from the first group.
 Factor out from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.
 Combine like terms. Or factor out the common term 
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So then factors further to 
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Answer:
So completely factors to .
In other words,  .
Note: you can check the answer by expanding to get or by graphing the original expression and the answer (the two graphs should be identical).
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Answer by Alan3354(69443)  (Show Source):
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