SOLUTION: how do you factor this problem? 2a^2 + 7a - 4.

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Question 193222: how do you factor this problem?
2a^2 + 7a - 4.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 2a%5E2%2B7a-4, we can see that the first coefficient is 2, the second coefficient is 7, and the last term is -4.


Now multiply the first coefficient 2 by the last term -4 to get %282%29%28-4%29=-8.


Now the question is: what two whole numbers multiply to -8 (the previous product) and add to the second coefficient 7?


To find these two numbers, we need to list all of the factors of -8 (the previous product).


Factors of -8:
1,2,4,8
-1,-2,-4,-8


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


These factors pair up and multiply to -8.
1*(-8)
2*(-4)
(-1)*(8)
(-2)*(4)

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 7:


First NumberSecond NumberSum
1-81+(-8)=-7
2-42+(-4)=-2
-18-1+8=7
-24-2+4=2



From the table, we can see that the two numbers -1 and 8 add to 7 (the middle coefficient).


So the two numbers -1 and 8 both multiply to -8 and add to 7


Now replace the middle term 7a with -a%2B8a. Remember, -1 and 8 add to 7. So this shows us that -a%2B8a=7a.


2a%5E2%2Bhighlight%28-a%2B8a%29-4 Replace the second term 7a with -a%2B8a.


%282a%5E2-a%29%2B%288a-4%29 Group the terms into two pairs.


a%282a-1%29%2B%288a-4%29 Factor out the GCF a from the first group.


a%282a-1%29%2B4%282a-1%29 Factor out 4 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.


%28a%2B4%29%282a-1%29 Combine like terms. Or factor out the common term 2a-1

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Answer:


So 2a%5E2%2B7a-4 factors to %28a%2B4%29%282a-1%29.


Note: you can check the answer by FOILing %28a%2B4%29%282a-1%29 to get 2a%5E2%2B7a-4 or by graphing the original expression and the answer (the two graphs should be identical).