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
| 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
So the original quadratic
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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Answer:
So the quadratic factors to
which can also be written as  since the factors repeat themselves
Notice how foils back to our original problem . This verifies our answer. |
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