SOLUTION: factor x^2-18x+81

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Question 560570: factor x^2-18x+81
Answer by jim_thompson5910(35256) About Me  (Show Source):
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Looking at the expression x%5E2-18x%2B81, we can see that the first coefficient is 1, the second coefficient is -18, and the last term is 81.


Now multiply the first coefficient 1 by the last term 81 to get %281%29%2881%29=81.


Now the question is: what two whole numbers multiply to 81 (the previous product) and add to the second coefficient -18?


To find these two numbers, we need to list all of the factors of 81 (the previous product).


Factors of 81:
1,3,9,27,81
-1,-3,-9,-27,-81


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


These factors pair up and multiply to 81.
1*81 = 81
3*27 = 81
9*9 = 81
(-1)*(-81) = 81
(-3)*(-27) = 81
(-9)*(-9) = 81

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -18:


First NumberSecond NumberSum
1811+81=82
3273+27=30
999+9=18
-1-81-1+(-81)=-82
-3-27-3+(-27)=-30
-9-9-9+(-9)=-18



From the table, we can see that the two numbers -9 and -9 add to -18 (the middle coefficient).


So the two numbers -9 and -9 both multiply to 81 and add to -18


Now replace the middle term -18x with -9x-9x. Remember, -9 and -9 add to -18. So this shows us that -9x-9x=-18x.


x%5E2%2Bhighlight%28-9x-9x%29%2B81 Replace the second term -18x with -9x-9x.


%28x%5E2-9x%29%2B%28-9x%2B81%29 Group the terms into two pairs.


x%28x-9%29%2B%28-9x%2B81%29 Factor out the GCF x from the first group.


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


%28x-9%29%5E2 Condense the terms.


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Answer:


So x%5E2-18x%2B81 factors to %28x-9%29%5E2.


In other words, x%5E2-18x%2B81=%28x-9%29%5E2.


Note: you can check the answer by expanding %28x-9%29%5E2 to get x%5E2-18x%2B81 or by graphing the original expression and the answer (the two graphs should be identical).
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