SOLUTION: Factor completely, if it is prime state this. -12x^2 – 28x + 24

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Question 463845: Factor completely, if it is prime state this. -12x^2 – 28x + 24
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


-12%2Ax%5E2-28%2Ax%2B24 Start with the given expression.



-4%283x%5E2%2B7x-6%29 Factor out the GCF -4.



Now let's try to factor the inner expression 3x%5E2%2B7x-6



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Looking at the expression 3x%5E2%2B7x-6, we can see that the first coefficient is 3, the second coefficient is 7, and the last term is -6.



Now multiply the first coefficient 3 by the last term -6 to get %283%29%28-6%29=-18.



Now the question is: what two whole numbers multiply to -18 (the previous product) and add to the second coefficient 7?



To find these two numbers, we need to list all of the factors of -18 (the previous product).



Factors of -18:

1,2,3,6,9,18

-1,-2,-3,-6,-9,-18



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



These factors pair up and multiply to -18.

1*(-18) = -18
2*(-9) = -18
3*(-6) = -18
(-1)*(18) = -18
(-2)*(9) = -18
(-3)*(6) = -18


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



First NumberSecond NumberSum
1-181+(-18)=-17
2-92+(-9)=-7
3-63+(-6)=-3
-118-1+18=17
-29-2+9=7
-36-3+6=3




From the table, we can see that the two numbers -2 and 9 add to 7 (the middle coefficient).



So the two numbers -2 and 9 both multiply to -18 and add to 7



Now replace the middle term 7x with -2x%2B9x. Remember, -2 and 9 add to 7. So this shows us that -2x%2B9x=7x.



3x%5E2%2Bhighlight%28-2x%2B9x%29-6 Replace the second term 7x with -2x%2B9x.



%283x%5E2-2x%29%2B%289x-6%29 Group the terms into two pairs.



x%283x-2%29%2B%289x-6%29 Factor out the GCF x from the first group.



x%283x-2%29%2B3%283x-2%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.



%28x%2B3%29%283x-2%29 Combine like terms. Or factor out the common term 3x-2



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So -4%283x%5E2%2B7x-6%29 then factors further to -4%28x%2B3%29%283x-2%29



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Answer:



So -12%2Ax%5E2-28%2Ax%2B24 completely factors to -4%28x%2B3%29%283x-2%29.



In other words, -12%2Ax%5E2-28%2Ax%2B24=-4%28x%2B3%29%283x-2%29.



Note: you can check the answer by expanding -4%28x%2B3%29%283x-2%29 to get -12%2Ax%5E2-28%2Ax%2B24 or by graphing the original expression and the answer (the two graphs should be identical).