SOLUTION: Factor completely. x2 – 11x + 18

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Question 117163: Factor completely. x2 – 11x + 18
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at x%5E2-11x%2B18 we can see that the first term is x%5E2 and the last term is 18 where the coefficients are 1 and 18 respectively.

Now multiply the first coefficient 1 and the last coefficient 18 to get 18. Now what two numbers multiply to 18 and add to the middle coefficient -11? Let's list all of the factors of 18:



Factors of 18:
1,2,3,6,9,18

-1,-2,-3,-6,-9,-18 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 18
1*18
2*9
3*6
(-1)*(-18)
(-2)*(-9)
(-3)*(-6)

note: remember two negative numbers multiplied together make a positive number


Now which of these pairs add to -11? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -11

First NumberSecond NumberSum
1181+18=19
292+9=11
363+6=9
-1-18-1+(-18)=-19
-2-9-2+(-9)=-11
-3-6-3+(-6)=-9



From this list we can see that -2 and -9 add up to -11 and multiply to 18


Now looking at the expression x%5E2-11x%2B18, replace -11x with -2x%2B-9x (notice -2x%2B-9x adds up to -11x. So it is equivalent to -11x)

x%5E2%2Bhighlight%28-2x%2B-9x%29%2B18


Now let's factor x%5E2-2x-9x%2B18 by grouping:


%28x%5E2-2x%29%2B%28-9x%2B18%29 Group like terms


x%28x-2%29-9%28x-2%29 Factor out the GCF of x out of the first group. Factor out the GCF of -9 out of the second group


%28x-9%29%28x-2%29 Since we have a common term of x-2, we can combine like terms

So x%5E2-2x-9x%2B18 factors to %28x-9%29%28x-2%29


So this also means that x%5E2-11x%2B18 factors to %28x-9%29%28x-2%29 (since x%5E2-11x%2B18 is equivalent to x%5E2-2x-9x%2B18)

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Answer:

So x%5E2-11x%2B18 factors to %28x-9%29%28x-2%29