SOLUTION: x^2-84x+83

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Question 704903: x^2-84x+83
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression x%5E2-84x%2B83, we can see that the first coefficient is 1, the second coefficient is -84, and the last term is 83.


Now multiply the first coefficient 1 by the last term 83 to get %281%29%2883%29=83.


Now the question is: what two whole numbers multiply to 83 (the previous product) and add to the second coefficient -84?


To find these two numbers, we need to list all of the factors of 83 (the previous product).


Factors of 83:
1,83
-1,-83


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


These factors pair up and multiply to 83.
1*83 = 83
(-1)*(-83) = 83

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -84:


First NumberSecond NumberSum
1831+83=84
-1-83-1+(-83)=-84



From the table, we can see that the two numbers -1 and -83 add to -84 (the middle coefficient).


So the two numbers -1 and -83 both multiply to 83 and add to -84


Now replace the middle term -84x with -x-83x. Remember, -1 and -83 add to -84. So this shows us that -x-83x=-84x.


x%5E2%2Bhighlight%28-x-83x%29%2B83 Replace the second term -84x with -x-83x.


%28x%5E2-x%29%2B%28-83x%2B83%29 Group the terms into two pairs.


x%28x-1%29%2B%28-83x%2B83%29 Factor out the GCF x from the first group.


x%28x-1%29-83%28x-1%29 Factor out 83 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-83%29%28x-1%29 Combine like terms. Or factor out the common term x-1


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Answer:


So x%5E2-84x%2B83 factors to %28x-83%29%28x-1%29.


In other words, x%5E2-84x%2B83=%28x-83%29%28x-1%29.


Note: you can check the answer by expanding %28x-83%29%28x-1%29 to get x%5E2-84x%2B83 or by graphing the original expression and the answer (the two graphs should be identical).