SOLUTION: What is the factored form of 12d squared + 4d - 1 ?

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Question 607717: What is the factored form of 12d squared + 4d - 1 ?
Answer by jim_thompson5910(35256) About Me  (Show Source):
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Looking at the expression 12d%5E2%2B4d-1, we can see that the first coefficient is 12, the second coefficient is 4, and the last term is -1.


Now multiply the first coefficient 12 by the last term -1 to get %2812%29%28-1%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 4d with -2d%2B6d. Remember, -2 and 6 add to 4. So this shows us that -2d%2B6d=4d.


12d%5E2%2Bhighlight%28-2d%2B6d%29-1 Replace the second term 4d with -2d%2B6d.


%2812d%5E2-2d%29%2B%286d-1%29 Group the terms into two pairs.


2d%286d-1%29%2B%286d-1%29 Factor out the GCF 2d from the first group.


2d%286d-1%29%2B1%286d-1%29 Factor out 1 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.


%282d%2B1%29%286d-1%29 Combine like terms. Or factor out the common term 6d-1


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Answer:


So 12d%5E2%2B4d-1 factors to %282d%2B1%29%286d-1%29.


In other words, 12d%5E2%2B4d-1=%282d%2B1%29%286d-1%29.


Note: you can check the answer by expanding %282d%2B1%29%286d-1%29 to get 12d%5E2%2B4d-1 or by graphing the original expression and the answer (the two graphs should be identical).

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