SOLUTION: how to factor 3w^2+7w-20

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Question 286517: how to factor 3w^2+7w-20
Answer by jim_thompson5910(35256) About Me  (Show Source):
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression 3w%5E2%2B7w-20, we can see that the first coefficient is 3, the second coefficient is 7, and the last term is -20.



Now multiply the first coefficient 3 by the last term -20 to get %283%29%28-20%29=-60.



Now the question is: what two whole numbers multiply to -60 (the previous product) and add to the second coefficient 7?



To find these two numbers, we need to list all of the factors of -60 (the previous product).



Factors of -60:

1,2,3,4,5,6,10,12,15,20,30,60

-1,-2,-3,-4,-5,-6,-10,-12,-15,-20,-30,-60



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



These factors pair up and multiply to -60.

1*(-60) = -60
2*(-30) = -60
3*(-20) = -60
4*(-15) = -60
5*(-12) = -60
6*(-10) = -60
(-1)*(60) = -60
(-2)*(30) = -60
(-3)*(20) = -60
(-4)*(15) = -60
(-5)*(12) = -60
(-6)*(10) = -60


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



First NumberSecond NumberSum
1-601+(-60)=-59
2-302+(-30)=-28
3-203+(-20)=-17
4-154+(-15)=-11
5-125+(-12)=-7
6-106+(-10)=-4
-160-1+60=59
-230-2+30=28
-320-3+20=17
-415-4+15=11
-512-5+12=7
-610-6+10=4




From the table, we can see that the two numbers -5 and 12 add to 7 (the middle coefficient).



So the two numbers -5 and 12 both multiply to -60 and add to 7



Now replace the middle term 7w with -5w%2B12w. Remember, -5 and 12 add to 7. So this shows us that -5w%2B12w=7w.



3w%5E2%2Bhighlight%28-5w%2B12w%29-20 Replace the second term 7w with -5w%2B12w.



%283w%5E2-5w%29%2B%2812w-20%29 Group the terms into two pairs.



w%283w-5%29%2B%2812w-20%29 Factor out the GCF w from the first group.



w%283w-5%29%2B4%283w-5%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.



%28w%2B4%29%283w-5%29 Combine like terms. Or factor out the common term 3w-5



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Answer:



So 3%2Aw%5E2%2B7%2Aw-20 factors to %28w%2B4%29%283w-5%29.



In other words, 3%2Aw%5E2%2B7%2Aw-20=%28w%2B4%29%283w-5%29.



Note: you can check the answer by expanding %28w%2B4%29%283w-5%29 to get 3%2Aw%5E2%2B7%2Aw-20 or by graphing the original expression and the answer (the two graphs should be identical).