# 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

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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)
In order to factor , first multiply the leading coefficient 64 and the last term 49 to get 3136. Now we need to ask ourselves: What two numbers multiply to 3136 and add to -112? Lets find out by listing all of the possible factors of 3136

Factors:

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, List the negative factors as well. This will allow us to find all possible combinations

These factors pair up to multiply to 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

(-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

note: remember two negative numbers multiplied together make a positive number

Now which of these pairs add to -112? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -112

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 First Number | Second Number | Sum 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 -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

We can see from the table that -56 and -56 add to -112. So the two numbers that multiply to 3136 and add to -112 are: -56 and -56

breaks down to this (just replace with the two numbers that multiply to 3136 and add to -112, which are: -56 and -56)

Replace with

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

Factor a 8b out of the first group and factor a -7 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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