SOLUTION: Rewrite the middle term as the sum of two terms and then factor by grouping x^2-14x+45

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Question 103934: Rewrite the middle term as the sum of two terms and then factor by grouping
x^2-14x+45
Found 2 solutions by JP, jim_thompson5910:
Answer by JP(22) About Me  (Show Source):
You can put this solution on YOUR website!
x^2-14x+45
-9x-5x=-14x, -9*-5=45, x*x=x^2
(x-9)(x-5) is the solution

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression x%5E2-14x%2B45, we can see that the first coefficient is 1, the second coefficient is -14, and the last term is 45.



Now multiply the first coefficient 1 by the last term 45 to get %281%29%2845%29=45.



Now the question is: what two whole numbers multiply to 45 (the previous product) and add to the second coefficient -14?



To find these two numbers, we need to list all of the factors of 45 (the previous product).



Factors of 45:

1,3,5,9,15,45

-1,-3,-5,-9,-15,-45



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



These factors pair up and multiply to 45.

1*45 = 45
3*15 = 45
5*9 = 45
(-1)*(-45) = 45
(-3)*(-15) = 45
(-5)*(-9) = 45


Now let's add up each pair of factors to see if one pair adds to the middle coefficient -14:



First NumberSecond NumberSum
1451+45=46
3153+15=18
595+9=14
-1-45-1+(-45)=-46
-3-15-3+(-15)=-18
-5-9-5+(-9)=-14




From the table, we can see that the two numbers -5 and -9 add to -14 (the middle coefficient).



So the two numbers -5 and -9 both multiply to 45 and add to -14



Now replace the middle term -14x with -5x-9x. Remember, -5 and -9 add to -14. So this shows us that -5x-9x=-14x.



x%5E2%2Bhighlight%28-5x-9x%29%2B45 Replace the second term -14x with -5x-9x.



%28x%5E2-5x%29%2B%28-9x%2B45%29 Group the terms into two pairs.



x%28x-5%29%2B%28-9x%2B45%29 Factor out the GCF x from the first group.



x%28x-5%29-9%28x-5%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-5%29 Combine like terms. Or factor out the common term x-5



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Answer:



So x%5E2-14%2Ax%2B45 factors to %28x-9%29%28x-5%29.



In other words, x%5E2-14%2Ax%2B45=%28x-9%29%28x-5%29.



Note: you can check the answer by expanding %28x-9%29%28x-5%29 to get x%5E2-14%2Ax%2B45 or by graphing the original expression and the answer (the two graphs should be identical).