SOLUTION: -80x^2-44x+180

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Question 578581: -80x^2-44x+180
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'm assuming you want to factor this.

-80x%5E2-44x%2B180 Start with the given expression.


-4%2820x%5E2%2B11x-45%29 Factor out the GCF -4.


Now let's try to factor the inner expression 20x%5E2%2B11x-45


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Looking at the expression 20x%5E2%2B11x-45, we can see that the first coefficient is 20, the second coefficient is 11, and the last term is -45.


Now multiply the first coefficient 20 by the last term -45 to get %2820%29%28-45%29=-900.


Now the question is: what two whole numbers multiply to -900 (the previous product) and add to the second coefficient 11?


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 11:


First NumberSecond NumberSum
1-9001+(-900)=-899
2-4502+(-450)=-448
3-3003+(-300)=-297
4-2254+(-225)=-221
5-1805+(-180)=-175
6-1506+(-150)=-144
9-1009+(-100)=-91
10-9010+(-90)=-80
12-7512+(-75)=-63
15-6015+(-60)=-45
18-5018+(-50)=-32
20-4520+(-45)=-25
25-3625+(-36)=-11
30-3030+(-30)=0
-1900-1+900=899
-2450-2+450=448
-3300-3+300=297
-4225-4+225=221
-5180-5+180=175
-6150-6+150=144
-9100-9+100=91
-1090-10+90=80
-1275-12+75=63
-1560-15+60=45
-1850-18+50=32
-2045-20+45=25
-2536-25+36=11
-3030-30+30=0



From the table, we can see that the two numbers -25 and 36 add to 11 (the middle coefficient).


So the two numbers -25 and 36 both multiply to -900 and add to 11


Now replace the middle term 11x with -25x%2B36x. Remember, -25 and 36 add to 11. So this shows us that -25x%2B36x=11x.


20x%5E2%2Bhighlight%28-25x%2B36x%29-45 Replace the second term 11x with -25x%2B36x.


%2820x%5E2-25x%29%2B%2836x-45%29 Group the terms into two pairs.


5x%284x-5%29%2B%2836x-45%29 Factor out the GCF 5x from the first group.


5x%284x-5%29%2B9%284x-5%29 Factor out 9 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%2B9%29%284x-5%29 Combine like terms. Or factor out the common term 4x-5


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So -4%2820x%5E2%2B11x-45%29 then factors further to -4%285x%2B9%29%284x-5%29


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Answer:


So -80x%5E2-44x%2B180 completely factors to -4%285x%2B9%29%284x-5%29.


In other words, -80x%5E2-44x%2B180=-4%285x%2B9%29%284x-5%29.


Note: you can check the answer by expanding -4%285x%2B9%29%284x-5%29 to get -80x%5E2-44x%2B180 or by graphing the original expression and the answer (the two graphs should be identical).