SOLUTION: Which of the following is equivalent to 6x 2 + x - 12 ? A. (2x + 3)(3x - 4) B. (2x - 3)(3x - 4) C. (2x + 3)(3x + 4) D. (6x + 3)(x - 12) E. (x + 3)(2x - 12) Please show me s

Algebra ->  Test  -> Lessons -> SOLUTION: Which of the following is equivalent to 6x 2 + x - 12 ? A. (2x + 3)(3x - 4) B. (2x - 3)(3x - 4) C. (2x + 3)(3x + 4) D. (6x + 3)(x - 12) E. (x + 3)(2x - 12) Please show me s      Log On


   

Question 281599: Which of the following is equivalent to 6x 2 + x - 12 ?
A. (2x + 3)(3x - 4)
B. (2x - 3)(3x - 4)
C. (2x + 3)(3x + 4)
D. (6x + 3)(x - 12)
E. (x + 3)(2x - 12)
Please show me step by step how to solve this problem
Thanks
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 6x%5E2%2Bx-12, we can see that the first coefficient is 6, the second coefficient is 1, and the last term is -12.


Now multiply the first coefficient 6 by the last term -12 to get %286%29%28-12%29=-72.


Now the question is: what two whole numbers multiply to -72 (the previous product) and add to the second coefficient 1?


To find these two numbers, we need to list all of the factors of -72 (the previous product).


Factors of -72:
1,2,3,4,6,8,9,12,18,24,36,72
-1,-2,-3,-4,-6,-8,-9,-12,-18,-24,-36,-72


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


These factors pair up and multiply to -72.
1*(-72) = -72
2*(-36) = -72
3*(-24) = -72
4*(-18) = -72
6*(-12) = -72
8*(-9) = -72
(-1)*(72) = -72
(-2)*(36) = -72
(-3)*(24) = -72
(-4)*(18) = -72
(-6)*(12) = -72
(-8)*(9) = -72

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 1:


First NumberSecond NumberSum
1-721+(-72)=-71
2-362+(-36)=-34
3-243+(-24)=-21
4-184+(-18)=-14
6-126+(-12)=-6
8-98+(-9)=-1
-172-1+72=71
-236-2+36=34
-324-3+24=21
-418-4+18=14
-612-6+12=6
-89-8+9=1



From the table, we can see that the two numbers -8 and 9 add to 1 (the middle coefficient).


So the two numbers -8 and 9 both multiply to -72 and add to 1


Now replace the middle term 1x with -8x%2B9x. Remember, -8 and 9 add to 1. So this shows us that -8x%2B9x=1x.


6x%5E2%2Bhighlight%28-8x%2B9x%29-12 Replace the second term 1x with -8x%2B9x.


%286x%5E2-8x%29%2B%289x-12%29 Group the terms into two pairs.


2x%283x-4%29%2B%289x-12%29 Factor out the GCF 2x from the first group.


2x%283x-4%29%2B3%283x-4%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%2B3%29%283x-4%29 Combine like terms. Or factor out the common term 3x-4


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Answer:


So 6x%5E2%2Bx-12 factors to %282x%2B3%29%283x-4%29.


In other words, 6x%5E2%2Bx-12=%282x%2B3%29%283x-4%29. So the answer is A)


Note: you can check the answer by expanding %282x%2B3%29%283x-4%29 to get 6x%5E2%2Bx-12 or by graphing the original expression and the answer (the two graphs should be identical).