SOLUTION: 8a^2+14a-49

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Question 672866: 8a^2+14a-49
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 8a%5E2%2B14a-49, we can see that the first coefficient is 8, the second coefficient is 14, and the last term is -49.


Now multiply the first coefficient 8 by the last term -49 to get %288%29%28-49%29=-392.


Now the question is: what two whole numbers multiply to -392 (the previous product) and add to the second coefficient 14?


To find these two numbers, we need to list all of the factors of -392 (the previous product).


Factors of -392:
1,2,4,7,8,14,28,49,56,98,196,392
-1,-2,-4,-7,-8,-14,-28,-49,-56,-98,-196,-392


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


These factors pair up and multiply to -392.
1*(-392) = -392
2*(-196) = -392
4*(-98) = -392
7*(-56) = -392
8*(-49) = -392
14*(-28) = -392
(-1)*(392) = -392
(-2)*(196) = -392
(-4)*(98) = -392
(-7)*(56) = -392
(-8)*(49) = -392
(-14)*(28) = -392

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 14:


First NumberSecond NumberSum
1-3921+(-392)=-391
2-1962+(-196)=-194
4-984+(-98)=-94
7-567+(-56)=-49
8-498+(-49)=-41
14-2814+(-28)=-14
-1392-1+392=391
-2196-2+196=194
-498-4+98=94
-756-7+56=49
-849-8+49=41
-1428-14+28=14



From the table, we can see that the two numbers -14 and 28 add to 14 (the middle coefficient).


So the two numbers -14 and 28 both multiply to -392 and add to 14


Now replace the middle term 14a with -14a%2B28a. Remember, -14 and 28 add to 14. So this shows us that -14a%2B28a=14a.


8a%5E2%2Bhighlight%28-14a%2B28a%29-49 Replace the second term 14a with -14a%2B28a.


%288a%5E2-14a%29%2B%2828a-49%29 Group the terms into two pairs.


2a%284a-7%29%2B%2828a-49%29 Factor out the GCF 2a from the first group.


2a%284a-7%29%2B7%284a-7%29 Factor out 7 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.


%282a%2B7%29%284a-7%29 Combine like terms. Or factor out the common term 4a-7


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Answer:


So 8a%5E2%2B14a-49 factors to %282a%2B7%29%284a-7%29.


In other words, 8a%5E2%2B14a-49=%282a%2B7%29%284a-7%29.


Note: you can check the answer by expanding %282a%2B7%29%284a-7%29 to get 8a%5E2%2B14a-49 or by graphing the original expression and the answer (the two graphs should be identical).