SOLUTION: how to factor completely: 16x^2 + 4x - 2 = ?

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Question 283992: how to factor completely: 16x^2 + 4x - 2 = ?
Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
16x^2 + 4x - 2
factor out 2
2*(8x^2+4x-1)
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression 8x%5E2%2B2x-1, we can see that the first coefficient is 8, the second coefficient is 2, and the last term is -1.



Now multiply the first coefficient 8 by the last term -1 to get %288%29%28-1%29=-8.



Now the question is: what two whole numbers multiply to -8 (the previous product) and add to the second coefficient 2?



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) = -8
2*(-4) = -8
(-1)*(8) = -8
(-2)*(4) = -8


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 2:



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 -2 and 4 add to 2 (the middle coefficient).



So the two numbers -2 and 4 both multiply to -8 and add to 2



Now replace the middle term 2x with -2x%2B4x. Remember, -2 and 4 add to 2. So this shows us that -2x%2B4x=2x.



8x%5E2%2Bhighlight%28-2x%2B4x%29-1 Replace the second term 2x with -2x%2B4x.



%288x%5E2-2x%29%2B%284x-1%29 Group the terms into two pairs.



2x%284x-1%29%2B%284x-1%29 Factor out the GCF 2x from the first group.



2x%284x-1%29%2B1%284x-1%29 Factor out 1 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.



%282x%2B1%29%284x-1%29 Combine like terms. Or factor out the common term 4x-1



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Answer:



So 8%2Ax%5E2%2B2%2Ax-1 factors to %282x%2B1%29%284x-1%29.



In other words, 8%2Ax%5E2%2B2%2Ax-1=%282x%2B1%29%284x-1%29.



Note: you can check the answer by expanding %282x%2B1%29%284x-1%29 to get 8%2Ax%5E2%2B2%2Ax-1 or by graphing the original expression and the answer (the two graphs should be identical).