SOLUTION: How do you solve the function 12b^2 - 7b - 12 by grouping?

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Question 500207: How do you solve the function 12b^2 - 7b - 12 by grouping?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 12b%5E2-7b-12, we can see that the first coefficient is 12, the second coefficient is -7, and the last term is -12.


Now multiply the first coefficient 12 by the last term -12 to get %2812%29%28-12%29=-144.


Now the question is: what two whole numbers multiply to -144 (the previous product) and add to the second coefficient -7?


To find these two numbers, we need to list all of the factors of -144 (the previous product).


Factors of -144:
1,2,3,4,6,8,9,12,16,18,24,36,48,72,144
-1,-2,-3,-4,-6,-8,-9,-12,-16,-18,-24,-36,-48,-72,-144


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


These factors pair up and multiply to -144.
1*(-144) = -144
2*(-72) = -144
3*(-48) = -144
4*(-36) = -144
6*(-24) = -144
8*(-18) = -144
9*(-16) = -144
12*(-12) = -144
(-1)*(144) = -144
(-2)*(72) = -144
(-3)*(48) = -144
(-4)*(36) = -144
(-6)*(24) = -144
(-8)*(18) = -144
(-9)*(16) = -144
(-12)*(12) = -144

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


First NumberSecond NumberSum
1-1441+(-144)=-143
2-722+(-72)=-70
3-483+(-48)=-45
4-364+(-36)=-32
6-246+(-24)=-18
8-188+(-18)=-10
9-169+(-16)=-7
12-1212+(-12)=0
-1144-1+144=143
-272-2+72=70
-348-3+48=45
-436-4+36=32
-624-6+24=18
-818-8+18=10
-916-9+16=7
-1212-12+12=0



From the table, we can see that the two numbers 9 and -16 add to -7 (the middle coefficient).


So the two numbers 9 and -16 both multiply to -144 and add to -7


Now replace the middle term -7b with 9b-16b. Remember, 9 and -16 add to -7. So this shows us that 9b-16b=-7b.


12b%5E2%2Bhighlight%289b-16b%29-12 Replace the second term -7b with 9b-16b.


%2812b%5E2%2B9b%29%2B%28-16b-12%29 Group the terms into two pairs.


3b%284b%2B3%29%2B%28-16b-12%29 Factor out the GCF 3b from the first group.


3b%284b%2B3%29-4%284b%2B3%29 Factor out 4 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.


%283b-4%29%284b%2B3%29 Combine like terms. Or factor out the common term 4b%2B3


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Answer:


So 12b%5E2-7b-12 factors to %283b-4%29%284b%2B3%29.


In other words, 12b%5E2-7b-12=%283b-4%29%284b%2B3%29.


Note: you can check the answer by expanding %283b-4%29%284b%2B3%29 to get 12b%5E2-7b-12 or by graphing the original expression and the answer (the two graphs should be identical).

If you need more help, feel free to email me at jim_thompson5910@hotmail.com

Thanks,

Jim