SOLUTION: Can someone help me solve this. I have to factor trinomials using the AC method. 20x^2-20x-15

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Question 110211: Can someone help me solve this. I have to factor trinomials using the AC method.
20x^2-20x-15
Found 2 solutions by jim_thompson5910, scott8148:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


20%2Ax%5E2-20%2Ax-15 Start with the given expression.



5%284x%5E2-4x-3%29 Factor out the GCF 5.



Now let's try to factor the inner expression 4x%5E2-4x-3



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Looking at the expression 4x%5E2-4x-3, we can see that the first coefficient is 4, the second coefficient is -4, and the last term is -3.



Now multiply the first coefficient 4 by the last term -3 to get %284%29%28-3%29=-12.



Now the question is: what two whole numbers multiply to -12 (the previous product) and add to the second coefficient -4?



To find these two numbers, we need to list all of the factors of -12 (the previous product).



Factors of -12:

1,2,3,4,6,12

-1,-2,-3,-4,-6,-12



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



These factors pair up and multiply to -12.

1*(-12) = -12
2*(-6) = -12
3*(-4) = -12
(-1)*(12) = -12
(-2)*(6) = -12
(-3)*(4) = -12


Now let's add up each pair of factors to see if one pair adds to the middle coefficient -4:



First NumberSecond NumberSum
1-121+(-12)=-11
2-62+(-6)=-4
3-43+(-4)=-1
-112-1+12=11
-26-2+6=4
-34-3+4=1




From the table, we can see that the two numbers 2 and -6 add to -4 (the middle coefficient).



So the two numbers 2 and -6 both multiply to -12 and add to -4



Now replace the middle term -4x with 2x-6x. Remember, 2 and -6 add to -4. So this shows us that 2x-6x=-4x.



4x%5E2%2Bhighlight%282x-6x%29-3 Replace the second term -4x with 2x-6x.



%284x%5E2%2B2x%29%2B%28-6x-3%29 Group the terms into two pairs.



2x%282x%2B1%29%2B%28-6x-3%29 Factor out the GCF 2x from the first group.



2x%282x%2B1%29-3%282x%2B1%29 Factor out 3 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-3%29%282x%2B1%29 Combine like terms. Or factor out the common term 2x%2B1



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So 5%284x%5E2-4x-3%29 then factors further to 5%282x-3%29%282x%2B1%29



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Answer:



So 20%2Ax%5E2-20%2Ax-15 completely factors to 5%282x-3%29%282x%2B1%29.



In other words, 20%2Ax%5E2-20%2Ax-15=5%282x-3%29%282x%2B1%29.



Note: you can check the answer by expanding 5%282x-3%29%282x%2B1%29 to get 20%2Ax%5E2-20%2Ax-15 or by graphing the original expression and the answer (the two graphs should be identical).

Answer by scott8148(6628) About Me  (Show Source):
You can put this solution on YOUR website!
5 is common to all the terms ...5(4x^2-4x-3)

AC means take the factors of the product of a and c, and try to add them to get b

4(-3)=-12 ... factors are 1, 2, 3, 4, 6, 12 ... 2 and -6 add to -4

(4x+2)(4x-6) ... reduce coefficients by common factors (2 in this case) ...(2x+1)(2x-3)

5(2x+1)(2x-3)