SOLUTION: Factor the expression 30 x^2 + 61 x + 30

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Question 902674: Factor the expression 30 x^2 + 61 x + 30
Answer by richwmiller(17219) About Me  (Show Source):
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression 30x%5E2%2B61x%2B30, we can see that the first coefficient is 30, the second coefficient is 61, and the last term is 30.



Now multiply the first coefficient 30 by the last term 30 to get %2830%29%2830%29=900.



Now the question is: what two whole numbers multiply to 900 (the previous product) and add to the second coefficient 61?



To find these two numbers, we need to list all of the factors of 900 (the previous product).



Factors of 900:

1,2,3,4,5,6,9,10,12,15,18,20,25,30,36,45,50,60,75,90,100,150,180,225,300,450,900

-1,-2,-3,-4,-5,-6,-9,-10,-12,-15,-18,-20,-25,-30,-36,-45,-50,-60,-75,-90,-100,-150,-180,-225,-300,-450,-900



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



These factors pair up and multiply to 900.

1*900 = 900
2*450 = 900
3*300 = 900
4*225 = 900
5*180 = 900
6*150 = 900
9*100 = 900
10*90 = 900
12*75 = 900
15*60 = 900
18*50 = 900
20*45 = 900
25*36 = 900
30*30 = 900
(-1)*(-900) = 900
(-2)*(-450) = 900
(-3)*(-300) = 900
(-4)*(-225) = 900
(-5)*(-180) = 900
(-6)*(-150) = 900
(-9)*(-100) = 900
(-10)*(-90) = 900
(-12)*(-75) = 900
(-15)*(-60) = 900
(-18)*(-50) = 900
(-20)*(-45) = 900
(-25)*(-36) = 900
(-30)*(-30) = 900


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 61:



First NumberSecond NumberSum
19001+900=901
24502+450=452
33003+300=303
42254+225=229
51805+180=185
61506+150=156
91009+100=109
109010+90=100
127512+75=87
156015+60=75
185018+50=68
204520+45=65
253625+36=61
303030+30=60
-1-900-1+(-900)=-901
-2-450-2+(-450)=-452
-3-300-3+(-300)=-303
-4-225-4+(-225)=-229
-5-180-5+(-180)=-185
-6-150-6+(-150)=-156
-9-100-9+(-100)=-109
-10-90-10+(-90)=-100
-12-75-12+(-75)=-87
-15-60-15+(-60)=-75
-18-50-18+(-50)=-68
-20-45-20+(-45)=-65
-25-36-25+(-36)=-61
-30-30-30+(-30)=-60




From the table, we can see that the two numbers 25 and 36 add to 61 (the middle coefficient).



So the two numbers 25 and 36 both multiply to 900 and add to 61



Now replace the middle term 61x with 25x%2B36x. Remember, 25 and 36 add to 61. So this shows us that 25x%2B36x=61x.



30x%5E2%2Bhighlight%2825x%2B36x%29%2B30 Replace the second term 61x with 25x%2B36x.



%2830x%5E2%2B25x%29%2B%2836x%2B30%29 Group the terms into two pairs.



5x%286x%2B5%29%2B%2836x%2B30%29 Factor out the GCF 5x from the first group.



5x%286x%2B5%29%2B6%286x%2B5%29 Factor out 6 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.



%285x%2B6%29%286x%2B5%29 Combine like terms. Or factor out the common term 6x%2B5



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Answer:



So 30%2Ax%5E2%2B61%2Ax%2B30 factors to %285x%2B6%29%286x%2B5%29.



In other words, 30%2Ax%5E2%2B61%2Ax%2B30=%285x%2B6%29%286x%2B5%29.



Note: you can check the answer by expanding %285x%2B6%29%286x%2B5%29 to get 30%2Ax%5E2%2B61%2Ax%2B30 or by graphing the original expression and the answer (the two graphs should be identical).