Question 116118: Can you please explain in detail how this problem is factored?
The direction say to factor completely.
24x^2 + 10x – 4
Thank you, all help is greatly appreciated.
Patti
Found 2 solutions by MathLover1, scott8148:
Answer by MathLover1(20850)   (Show Source):
You can put this solution on YOUR website!
| Solved by pluggable solver: Factoring using the AC method (Factor by Grouping) |
 Start with the given expression.
 Factor out the GCF  .
Now let's try to factor the inner expression 
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Looking at the expression , we can see that the first coefficient is  , the second coefficient is  , and the last term is  .
Now multiply the first coefficient by the last term to get  .
Now the question is: what two whole numbers multiply to  (the previous product) and add to the second coefficient ?
To find these two numbers, we need to list all of the factors of (the previous product).
Factors of :
1,2,3,4,6,8,12,24
-1,-2,-3,-4,-6,-8,-12,-24
Note: list the negative of each factor. This will allow us to find all possible combinations.
These factors pair up and multiply to .
1*(-24) = -24
2*(-12) = -24
3*(-8) = -24
4*(-6) = -24
(-1)*(24) = -24
(-2)*(12) = -24
(-3)*(8) = -24
(-4)*(6) = -24
Now let's add up each pair of factors to see if one pair adds to the middle coefficient :
| First Number | Second Number | Sum | | 1 | -24 | 1+(-24)=-23 | | 2 | -12 | 2+(-12)=-10 | | 3 | -8 | 3+(-8)=-5 | | 4 | -6 | 4+(-6)=-2 | | -1 | 24 | -1+24=23 | | -2 | 12 | -2+12=10 | | -3 | 8 | -3+8=5 | | -4 | 6 | -4+6=2 |
From the table, we can see that the two numbers  and  add to (the middle coefficient).
So the two numbers and both multiply to and add to
Now replace the middle term  with  . Remember, and add to . So this shows us that  .
 Replace the second term with .
 Group the terms into two pairs.
 Factor out the GCF  from the first group.
 Factor out 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.
 Combine like terms. Or factor out the common term 
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So then factors further to 
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Answer:
So completely factors to .
In other words,  .
Note: you can check the answer by expanding to get or by graphing the original expression and the answer (the two graphs should be identical).
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Answer by scott8148(6628)  (Show Source):
You can put this solution on YOUR website! 2 is a common factor for all the terms, so that is factored out first
2(12x^2+5x-2) ___ now it gets interesting
the factors of "a" (12) have to combine (multiply and add) with the factors of "c" (-2) to get "b" (5)
factors can be positive or negative
factors of "a" are 1 & 12; 2 & 6; 3 & 4 ___ either both positive or both negative
factors of "c" are 1 & 2 ___ one is positive, the other is negative
2 times 4 is 8; -1 times 3 is -3; 8 plus -3 is 5
2(4x-1)(3x+2)
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