SOLUTION: Need help with finding the factor for the perfect square trinomial 64b to the second-112b+49 thanks

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Question 264016: Need help with finding the factor for the perfect square trinomial
64b to the second-112b+49 thanks
Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


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 :

1,2,4,7,8,14,16,28,32,49,56,64,98,112,196,224,392,448,784,1568,3136

-1,-2,-4,-7,-8,-14,-16,-28,-32,-49,-56,-64,-98,-112,-196,-224,-392,-448,-784,-1568,-3136



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to .

1*3136 = 3136
2*1568 = 3136
4*784 = 3136
7*448 = 3136
8*392 = 3136
14*224 = 3136
16*196 = 3136
28*112 = 3136
32*98 = 3136
49*64 = 3136
56*56 = 3136
(-1)*(-3136) = 3136
(-2)*(-1568) = 3136
(-4)*(-784) = 3136
(-7)*(-448) = 3136
(-8)*(-392) = 3136
(-14)*(-224) = 3136
(-16)*(-196) = 3136
(-28)*(-112) = 3136
(-32)*(-98) = 3136
(-49)*(-64) = 3136
(-56)*(-56) = 3136


Now let's add up each pair of factors to see if one pair adds to the middle coefficient :



First NumberSecond NumberSum
131361+3136=3137
215682+1568=1570
47844+784=788
74487+448=455
83928+392=400
1422414+224=238
1619616+196=212
2811228+112=140
329832+98=130
496449+64=113
565656+56=112
-1-3136-1+(-3136)=-3137
-2-1568-2+(-1568)=-1570
-4-784-4+(-784)=-788
-7-448-7+(-448)=-455
-8-392-8+(-392)=-400
-14-224-14+(-224)=-238
-16-196-16+(-196)=-212
-28-112-28+(-112)=-140
-32-98-32+(-98)=-130
-49-64-49+(-64)=-113
-56-56-56+(-56)=-112




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



Condense the terms.



===============================================================



Answer:



So 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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