SOLUTION: I've been trying to factor this all night but cannot figure it out. Please help...16z^2-54z+35 I've tried AC=560, factors 40 and 14 but that doesn't account for the -54, it's pos

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Question 714299: I've been trying to factor this all night but cannot figure it out. Please help...16z^2-54z+35 I've tried AC=560, factors 40 and 14 but that doesn't account for the -54, it's positive 54.
Found 2 solutions by jim_thompson5910, solver91311:
Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!

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 :
1,2,4,5,7,8,10,14,16,20,28,35,40,56,70,80,112,140,280,560
-1,-2,-4,-5,-7,-8,-10,-14,-16,-20,-28,-35,-40,-56,-70,-80,-112,-140,-280,-560


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


These factors pair up and multiply to .
1*560 = 560
2*280 = 560
4*140 = 560
5*112 = 560
7*80 = 560
8*70 = 560
10*56 = 560
14*40 = 560
16*35 = 560
20*28 = 560
(-1)*(-560) = 560
(-2)*(-280) = 560
(-4)*(-140) = 560
(-5)*(-112) = 560
(-7)*(-80) = 560
(-8)*(-70) = 560
(-10)*(-56) = 560
(-14)*(-40) = 560
(-16)*(-35) = 560
(-20)*(-28) = 560

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


First NumberSecond NumberSum
15601+560=561
22802+280=282
41404+140=144
51125+112=117
7807+80=87
8708+70=78
105610+56=66
144014+40=54
163516+35=51
202820+28=48
-1-560-1+(-560)=-561
-2-280-2+(-280)=-282
-4-140-4+(-140)=-144
-5-112-5+(-112)=-117
-7-80-7+(-80)=-87
-8-70-8+(-70)=-78
-10-56-10+(-56)=-66
-14-40-14+(-40)=-54
-16-35-16+(-35)=-51
-20-28-20+(-28)=-48



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).
Answer by solver91311(24713)   (Show Source): You can put this solution on YOUR website!


Hint: 2 times 8 is 16 and -5 times -7 is 35

John

Egw to Beta kai to Sigma
My calculator said it, I believe it, that settles it
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