SOLUTION: 80a2-26a-15

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Question 668046: 80a2-26a-15
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
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Looking at the expression 80a%5E2-26a-15, we can see that the first coefficient is 80, the second coefficient is -26, and the last term is -15.


Now multiply the first coefficient 80 by the last term -15 to get %2880%29%28-15%29=-1200.


Now the question is: what two whole numbers multiply to -1200 (the previous product) and add to the second coefficient -26?


To find these two numbers, we need to list all of the factors of -1200 (the previous product).


Factors of -1200:
1,2,3,4,5,6,8,10,12,15,16,20,24,25,30,40,48,50,60,75,80,100,120,150,200,240,300,400,600,1200
-1,-2,-3,-4,-5,-6,-8,-10,-12,-15,-16,-20,-24,-25,-30,-40,-48,-50,-60,-75,-80,-100,-120,-150,-200,-240,-300,-400,-600,-1200


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


These factors pair up and multiply to -1200.
1*(-1200) = -1200
2*(-600) = -1200
3*(-400) = -1200
4*(-300) = -1200
5*(-240) = -1200
6*(-200) = -1200
8*(-150) = -1200
10*(-120) = -1200
12*(-100) = -1200
15*(-80) = -1200
16*(-75) = -1200
20*(-60) = -1200
24*(-50) = -1200
25*(-48) = -1200
30*(-40) = -1200
(-1)*(1200) = -1200
(-2)*(600) = -1200
(-3)*(400) = -1200
(-4)*(300) = -1200
(-5)*(240) = -1200
(-6)*(200) = -1200
(-8)*(150) = -1200
(-10)*(120) = -1200
(-12)*(100) = -1200
(-15)*(80) = -1200
(-16)*(75) = -1200
(-20)*(60) = -1200
(-24)*(50) = -1200
(-25)*(48) = -1200
(-30)*(40) = -1200

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -26:


First NumberSecond NumberSum
1-12001+(-1200)=-1199
2-6002+(-600)=-598
3-4003+(-400)=-397
4-3004+(-300)=-296
5-2405+(-240)=-235
6-2006+(-200)=-194
8-1508+(-150)=-142
10-12010+(-120)=-110
12-10012+(-100)=-88
15-8015+(-80)=-65
16-7516+(-75)=-59
20-6020+(-60)=-40
24-5024+(-50)=-26
25-4825+(-48)=-23
30-4030+(-40)=-10
-11200-1+1200=1199
-2600-2+600=598
-3400-3+400=397
-4300-4+300=296
-5240-5+240=235
-6200-6+200=194
-8150-8+150=142
-10120-10+120=110
-12100-12+100=88
-1580-15+80=65
-1675-16+75=59
-2060-20+60=40
-2450-24+50=26
-2548-25+48=23
-3040-30+40=10



From the table, we can see that the two numbers 24 and -50 add to -26 (the middle coefficient).


So the two numbers 24 and -50 both multiply to -1200 and add to -26


Now replace the middle term -26a with 24a-50a. Remember, 24 and -50 add to -26. So this shows us that 24a-50a=-26a.


80a%5E2%2Bhighlight%2824a-50a%29-15 Replace the second term -26a with 24a-50a.


%2880a%5E2%2B24a%29%2B%28-50a-15%29 Group the terms into two pairs.


8a%2810a%2B3%29%2B%28-50a-15%29 Factor out the GCF 8a from the first group.


8a%2810a%2B3%29-5%2810a%2B3%29 Factor out 5 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.


%288a-5%29%2810a%2B3%29 Combine like terms. Or factor out the common term 10a%2B3


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Answer:


So 80a%5E2-26a-15 factors to %288a-5%29%2810a%2B3%29.


In other words, 80a%5E2-26a-15=%288a-5%29%2810a%2B3%29.


Note: you can check the answer by expanding %288a-5%29%2810a%2B3%29 to get 80a%5E2-26a-15 or by graphing the original expression and the answer (the two graphs should be identical).