SOLUTION: Basic factoring of x^2+2x-48

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Question 714312: Basic factoring of x^2+2x-48
Found 2 solutions by Alan3354, jim_thompson5910:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Basic factoring of x^2+2x-48
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Find a pair of factors of 48 that differ by 2

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression x%5E2%2B2x-48, we can see that the first coefficient is 1, the second coefficient is 2, and the last term is -48.


Now multiply the first coefficient 1 by the last term -48 to get %281%29%28-48%29=-48.


Now the question is: what two whole numbers multiply to -48 (the previous product) and add to the second coefficient 2?


To find these two numbers, we need to list all of the factors of -48 (the previous product).


Factors of -48:
1,2,3,4,6,8,12,16,24,48
-1,-2,-3,-4,-6,-8,-12,-16,-24,-48


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


These factors pair up and multiply to -48.
1*(-48) = -48
2*(-24) = -48
3*(-16) = -48
4*(-12) = -48
6*(-8) = -48
(-1)*(48) = -48
(-2)*(24) = -48
(-3)*(16) = -48
(-4)*(12) = -48
(-6)*(8) = -48

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 2:


First NumberSecond NumberSum
1-481+(-48)=-47
2-242+(-24)=-22
3-163+(-16)=-13
4-124+(-12)=-8
6-86+(-8)=-2
-148-1+48=47
-224-2+24=22
-316-3+16=13
-412-4+12=8
-68-6+8=2



From the table, we can see that the two numbers -6 and 8 add to 2 (the middle coefficient).


So the two numbers -6 and 8 both multiply to -48 and add to 2


Now replace the middle term 2x with -6x%2B8x. Remember, -6 and 8 add to 2. So this shows us that -6x%2B8x=2x.


x%5E2%2Bhighlight%28-6x%2B8x%29-48 Replace the second term 2x with -6x%2B8x.


%28x%5E2-6x%29%2B%288x-48%29 Group the terms into two pairs.


x%28x-6%29%2B%288x-48%29 Factor out the GCF x from the first group.


x%28x-6%29%2B8%28x-6%29 Factor out 8 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%2B8%29%28x-6%29 Combine like terms. Or factor out the common term x-6


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Answer:


So x%5E2%2B2x-48 factors to %28x%2B8%29%28x-6%29.


In other words, x%5E2%2B2x-48=%28x%2B8%29%28x-6%29.


Note: you can check the answer by expanding %28x%2B8%29%28x-6%29 to get x%5E2%2B2x-48 or by graphing the original expression and the answer (the two graphs should be identical).