SOLUTION: 8x^2-73x+9

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Question 565148: 8x^2-73x+9
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.




Looking at the expression 8x%5E2-73x%2B9, we can see that the first coefficient is 8, the second coefficient is -73, and the last term is 9.


Now multiply the first coefficient 8 by the last term 9 to get %288%29%289%29=72.


Now the question is: what two whole numbers multiply to 72 (the previous product) and add to the second coefficient -73?


To find these two numbers, we need to list all of the factors of 72 (the previous product).


Factors of 72:
1,2,3,4,6,8,9,12,18,24,36,72
-1,-2,-3,-4,-6,-8,-9,-12,-18,-24,-36,-72


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


These factors pair up and multiply to 72.
1*72 = 72
2*36 = 72
3*24 = 72
4*18 = 72
6*12 = 72
8*9 = 72
(-1)*(-72) = 72
(-2)*(-36) = 72
(-3)*(-24) = 72
(-4)*(-18) = 72
(-6)*(-12) = 72
(-8)*(-9) = 72

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -73:


First NumberSecond NumberSum
1721+72=73
2362+36=38
3243+24=27
4184+18=22
6126+12=18
898+9=17
-1-72-1+(-72)=-73
-2-36-2+(-36)=-38
-3-24-3+(-24)=-27
-4-18-4+(-18)=-22
-6-12-6+(-12)=-18
-8-9-8+(-9)=-17



From the table, we can see that the two numbers -1 and -72 add to -73 (the middle coefficient).


So the two numbers -1 and -72 both multiply to 72 and add to -73


Now replace the middle term -73x with -x-72x. Remember, -1 and -72 add to -73. So this shows us that -x-72x=-73x.


8x%5E2%2Bhighlight%28-x-72x%29%2B9 Replace the second term -73x with -x-72x.


%288x%5E2-x%29%2B%28-72x%2B9%29 Group the terms into two pairs.


x%288x-1%29%2B%28-72x%2B9%29 Factor out the GCF x from the first group.


x%288x-1%29-9%288x-1%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.


%28x-9%29%288x-1%29 Combine like terms. Or factor out the common term 8x-1


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Answer:


So 8x%5E2-73x%2B9 factors to %28x-9%29%288x-1%29.


In other words, 8x%5E2-73x%2B9=%28x-9%29%288x-1%29.


Note: you can check the answer by expanding %28x-9%29%288x-1%29 to get 8x%5E2-73x%2B9 or by graphing the original expression and the answer (the two graphs should be identical).
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