SOLUTION: factor x^2-9x+14

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Question 560571: factor x^2-9x+14
Answer by jim_thompson5910(35256) About Me  (Show Source):
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Looking at the expression x%5E2-9x%2B14, we can see that the first coefficient is 1, the second coefficient is -9, and the last term is 14.


Now multiply the first coefficient 1 by the last term 14 to get %281%29%2814%29=14.


Now the question is: what two whole numbers multiply to 14 (the previous product) and add to the second coefficient -9?


To find these two numbers, we need to list all of the factors of 14 (the previous product).


Factors of 14:
1,2,7,14
-1,-2,-7,-14


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


These factors pair up and multiply to 14.
1*14 = 14
2*7 = 14
(-1)*(-14) = 14
(-2)*(-7) = 14

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -9:


First NumberSecond NumberSum
1141+14=15
272+7=9
-1-14-1+(-14)=-15
-2-7-2+(-7)=-9



From the table, we can see that the two numbers -2 and -7 add to -9 (the middle coefficient).


So the two numbers -2 and -7 both multiply to 14 and add to -9


Now replace the middle term -9x with -2x-7x. Remember, -2 and -7 add to -9. So this shows us that -2x-7x=-9x.


x%5E2%2Bhighlight%28-2x-7x%29%2B14 Replace the second term -9x with -2x-7x.


%28x%5E2-2x%29%2B%28-7x%2B14%29 Group the terms into two pairs.


x%28x-2%29%2B%28-7x%2B14%29 Factor out the GCF x from the first group.


x%28x-2%29-7%28x-2%29 Factor out 7 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-7%29%28x-2%29 Combine like terms. Or factor out the common term x-2


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Answer:


So x%5E2-9x%2B14 factors to %28x-7%29%28x-2%29.


In other words, x%5E2-9x%2B14=%28x-7%29%28x-2%29.


Note: you can check the answer by expanding %28x-7%29%28x-2%29 to get x%5E2-9x%2B14 or by graphing the original expression and the answer (the two graphs should be identical).
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