SOLUTION: can you help me solve the equation by factoring x^2-21x+108=0

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Question 609785: can you help me solve the equation by factoring
x^2-21x+108=0
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'll help you factor this. Once the factorization is done, I'll let you take over.


Looking at the expression x%5E2-21x%2B108, we can see that the first coefficient is 1, the second coefficient is -21, and the last term is 108.


Now multiply the first coefficient 1 by the last term 108 to get %281%29%28108%29=108.


Now the question is: what two whole numbers multiply to 108 (the previous product) and add to the second coefficient -21?


To find these two numbers, we need to list all of the factors of 108 (the previous product).


Factors of 108:
1,2,3,4,6,9,12,18,27,36,54,108
-1,-2,-3,-4,-6,-9,-12,-18,-27,-36,-54,-108


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


These factors pair up and multiply to 108.
1*108 = 108
2*54 = 108
3*36 = 108
4*27 = 108
6*18 = 108
9*12 = 108
(-1)*(-108) = 108
(-2)*(-54) = 108
(-3)*(-36) = 108
(-4)*(-27) = 108
(-6)*(-18) = 108
(-9)*(-12) = 108

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -21:


First NumberSecond NumberSum
11081+108=109
2542+54=56
3363+36=39
4274+27=31
6186+18=24
9129+12=21
-1-108-1+(-108)=-109
-2-54-2+(-54)=-56
-3-36-3+(-36)=-39
-4-27-4+(-27)=-31
-6-18-6+(-18)=-24
-9-12-9+(-12)=-21



From the table, we can see that the two numbers -9 and -12 add to -21 (the middle coefficient).


So the two numbers -9 and -12 both multiply to 108 and add to -21


Now replace the middle term -21x with -9x-12x. Remember, -9 and -12 add to -21. So this shows us that -9x-12x=-21x.


x%5E2%2Bhighlight%28-9x-12x%29%2B108 Replace the second term -21x with -9x-12x.


%28x%5E2-9x%29%2B%28-12x%2B108%29 Group the terms into two pairs.


x%28x-9%29%2B%28-12x%2B108%29 Factor out the GCF x from the first group.


x%28x-9%29-12%28x-9%29 Factor out 12 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-12%29%28x-9%29 Combine like terms. Or factor out the common term x-9


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Answer:


So x%5E2-21x%2B108 factors to %28x-12%29%28x-9%29.


In other words, x%5E2-21x%2B108=%28x-12%29%28x-9%29.


Note: you can check the answer by expanding %28x-12%29%28x-9%29 to get x%5E2-21x%2B108 or by graphing the original expression and the answer (the two graphs should be identical).


So this means x%5E2-21x%2B108=0 turns into %28x-12%29%28x-9%29=0
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