SOLUTION: There was not a topic for what I am looking for. My problem says factor the trinomial completely and I'm not sure how to do it. 54t^2+3t-70 +(?)(?)

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Question 207384: There was not a topic for what I am looking for. My problem says factor the trinomial completely and I'm not sure how to do it.
54t^2+3t-70
+(?)(?)
Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!


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Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last term is .


Now multiply the first coefficient by the last term to get .


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


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


Factors of :
1,2,3,4,5,6,7,9,10,12,14,15,18,20,21,27,28,30,35,36,42,45,54,60,63,70,84,90,105,108,126,135,140,180,189,210,252,270,315,378,420,540,630,756,945,1260,1890,3780
-1,-2,-3,-4,-5,-6,-7,-9,-10,-12,-14,-15,-18,-20,-21,-27,-28,-30,-35,-36,-42,-45,-54,-60,-63,-70,-84,-90,-105,-108,-126,-135,-140,-180,-189,-210,-252,-270,-315,-378,-420,-540,-630,-756,-945,-1260,-1890,-3780


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


These factors pair up and multiply to .
1*(-3780) = -3780
2*(-1890) = -3780
3*(-1260) = -3780
4*(-945) = -3780
5*(-756) = -3780
6*(-630) = -3780
7*(-540) = -3780
9*(-420) = -3780
10*(-378) = -3780
12*(-315) = -3780
14*(-270) = -3780
15*(-252) = -3780
18*(-210) = -3780
20*(-189) = -3780
21*(-180) = -3780
27*(-140) = -3780
28*(-135) = -3780
30*(-126) = -3780
35*(-108) = -3780
36*(-105) = -3780
42*(-90) = -3780
45*(-84) = -3780
54*(-70) = -3780
60*(-63) = -3780
(-1)*(3780) = -3780
(-2)*(1890) = -3780
(-3)*(1260) = -3780
(-4)*(945) = -3780
(-5)*(756) = -3780
(-6)*(630) = -3780
(-7)*(540) = -3780
(-9)*(420) = -3780
(-10)*(378) = -3780
(-12)*(315) = -3780
(-14)*(270) = -3780
(-15)*(252) = -3780
(-18)*(210) = -3780
(-20)*(189) = -3780
(-21)*(180) = -3780
(-27)*(140) = -3780
(-28)*(135) = -3780
(-30)*(126) = -3780
(-35)*(108) = -3780
(-36)*(105) = -3780
(-42)*(90) = -3780
(-45)*(84) = -3780
(-54)*(70) = -3780
(-60)*(63) = -3780

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


First NumberSecond NumberSum
1-37801+(-3780)=-3779
2-18902+(-1890)=-1888
3-12603+(-1260)=-1257
4-9454+(-945)=-941
5-7565+(-756)=-751
6-6306+(-630)=-624
7-5407+(-540)=-533
9-4209+(-420)=-411
10-37810+(-378)=-368
12-31512+(-315)=-303
14-27014+(-270)=-256
15-25215+(-252)=-237
18-21018+(-210)=-192
20-18920+(-189)=-169
21-18021+(-180)=-159
27-14027+(-140)=-113
28-13528+(-135)=-107
30-12630+(-126)=-96
35-10835+(-108)=-73
36-10536+(-105)=-69
42-9042+(-90)=-48
45-8445+(-84)=-39
54-7054+(-70)=-16
60-6360+(-63)=-3
-13780-1+3780=3779
-21890-2+1890=1888
-31260-3+1260=1257
-4945-4+945=941
-5756-5+756=751
-6630-6+630=624
-7540-7+540=533
-9420-9+420=411
-10378-10+378=368
-12315-12+315=303
-14270-14+270=256
-15252-15+252=237
-18210-18+210=192
-20189-20+189=169
-21180-21+180=159
-27140-27+140=113
-28135-28+135=107
-30126-30+126=96
-35108-35+108=73
-36105-36+105=69
-4290-42+90=48
-4584-45+84=39
-5470-54+70=16
-6063-60+63=3



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


So the two numbers and both multiply to and add to


Now replace the middle term with . Remember, and add to . So this shows us that .


Replace the second term with .


Group the terms into two pairs.


Factor out the GCF from the first group.


Factor out 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.


Combine like terms. Or factor out the common term


===============================================================



Answer:


So factors to .


In other words, .


Note: you can check the answer by expanding to get or by graphing the original expression and the answer (the two graphs should be identical).

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