SOLUTION: Factor the trinomial 42b^2-43b-7=

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Question 179598: Factor the trinomial 42b^2-43b-7=
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 42b%5E2-43b-7, we can see that the first coefficient is 42, the second coefficient is -43, and the last term is -7.


Now multiply the first coefficient 42 by the last term -7 to get %2842%29%28-7%29=-294.


Now the question is: what two whole numbers multiply to -294 (the previous product) and add to the second coefficient -43?


To find these two numbers, we need to list all of the factors of -294 (the previous product).


Factors of -294:
1,2,3,6,7,14,21,42,49,98,147,294
-1,-2,-3,-6,-7,-14,-21,-42,-49,-98,-147,-294


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


These factors pair up and multiply to -294.
1*(-294)
2*(-147)
3*(-98)
6*(-49)
7*(-42)
14*(-21)
(-1)*(294)
(-2)*(147)
(-3)*(98)
(-6)*(49)
(-7)*(42)
(-14)*(21)

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -43:


First NumberSecond NumberSum
1-2941+(-294)=-293
2-1472+(-147)=-145
3-983+(-98)=-95
6-496+(-49)=-43
7-427+(-42)=-35
14-2114+(-21)=-7
-1294-1+294=293
-2147-2+147=145
-398-3+98=95
-649-6+49=43
-742-7+42=35
-1421-14+21=7



From the table, we can see that the two numbers 6 and -49 add to -43 (the middle coefficient).


So the two numbers 6 and -49 both multiply to -294 and add to -43


Now replace the middle term -43b with 6b-49b. Remember, 6 and -49 add to -43. So this shows us that 6b-49b=-43b.


42b%5E2%2Bhighlight%286b-49b%29-7 Replace the second term -43b with 6b-49b.


%2842b%5E2%2B6b%29%2B%28-49b-7%29 Group the terms into two pairs.


6b%287b%2B1%29%2B%28-49b-7%29 Factor out the GCF 6b from the first group.


6b%287b%2B1%29-7%287b%2B1%29 Factor out 7 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.


%286b-7%29%287b%2B1%29 Combine like terms. Or factor out the common term 7b%2B1

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Answer:


So 42b%5E2-43b-7 factors to %286b-7%29%287b%2B1%29.


Note: you can check the answer by FOILing %286b-7%29%287b%2B1%29 to get 42b%5E2-43b-7 or by graphing the original expression and the answer (the two graphs should be identical).