Question 132424: Factor.
a^2 + 16a + 64
Found 2 solutions by jim_thompson5910, edjones:
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
Looking at  we can see that the first term is  and the last term is  where the coefficients are 1 and 64 respectively.
Now multiply the first coefficient 1 and the last coefficient 64 to get 64. Now what two numbers multiply to 64 and add to the middle coefficient 16? Let's list all of the factors of 64:
Factors of 64:
1,2,4,8,16,32
-1,-2,-4,-8,-16,-32 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to 64
1*64
2*32
4*16
8*8
(-1)*(-64)
(-2)*(-32)
(-4)*(-16)
(-8)*(-8)
note: remember two negative numbers multiplied together make a positive number
Now which of these pairs add to 16? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 16
| First Number | Second Number | Sum | | 1 | 64 | 1+64=65 | | 2 | 32 | 2+32=34 | | 4 | 16 | 4+16=20 | | 8 | 8 | 8+8=16 | | -1 | -64 | -1+(-64)=-65 | | -2 | -32 | -2+(-32)=-34 | | -4 | -16 | -4+(-16)=-20 | | -8 | -8 | -8+(-8)=-16 |
From this list we can see that 8 and 8 add up to 16 and multiply to 64
Now looking at the expression , replace  with  (notice adds up to . So it is equivalent to )
Now let's factor  by grouping:
 Group like terms
 Factor out the GCF of  out of the first group. Factor out the GCF of  out of the second group
 Since we have a common term of  , we can combine like terms
So factors to
So this also means that factors to (since is equivalent to )
note: is equivalent to  since the term occurs twice. So also factors to
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Answer:
So  factors to
Answer by edjones(8007)  (Show Source):
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