SOLUTION: I am having trouble figuring out how to factor the three problems listed below:1. x^2+2x-2x-4;2.v^2+9v+14;3. r^2-2r-48. I have been working on them for about 1 1/2 hours at this ti

Algebra ->  Equations -> SOLUTION: I am having trouble figuring out how to factor the three problems listed below:1. x^2+2x-2x-4;2.v^2+9v+14;3. r^2-2r-48. I have been working on them for about 1 1/2 hours at this ti      Log On


   

Question 317097: I am having trouble figuring out how to factor the three problems listed below:1. x^2+2x-2x-4;2.v^2+9v+14;3. r^2-2r-48. I have been working on them for about 1 1/2 hours at this time and just can not seem to get any where can you help?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
# 1

x%5E2%2B2x-2x-4 Start with the given expression


%28x%5E2%2B2x%29%2B%28-2x-4%29 Group like terms


x%28x%2B2%29-2%28x%2B2%29 Factor out the GCF x out of the first group. Factor out the GCF -2 out of the second group


%28x-2%29%28x%2B2%29 Since we have the common term x%2B2, we can combine like terms


So x%5E2%2B2x-2x-4 factors to %28x-2%29%28x%2B2%29


In other words, x%5E2%2B2x-2x-4=%28x-2%29%28x%2B2%29


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# 2



Looking at the expression v%5E2%2B9v%2B14, we can see that the first coefficient is 1, the second coefficient is 9, and the last term is 14.


Now multiply the first coefficient 1 by the last term 14 to get %281%29%2814%29=14.


Now the question is: what two whole numbers multiply to 14 (the previous product) and add to the second coefficient 9?


To find these two numbers, we need to list all of the factors of 14 (the previous product).


Factors of 14:
1,2,7,14
-1,-2,-7,-14


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


These factors pair up and multiply to 14.
1*14 = 14
2*7 = 14
(-1)*(-14) = 14
(-2)*(-7) = 14

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 9:


First NumberSecond NumberSum
1141+14=15
272+7=9
-1-14-1+(-14)=-15
-2-7-2+(-7)=-9



From the table, we can see that the two numbers 2 and 7 add to 9 (the middle coefficient).


So the two numbers 2 and 7 both multiply to 14 and add to 9


Now replace the middle term 9v with 2v%2B7v. Remember, 2 and 7 add to 9. So this shows us that 2v%2B7v=9v.


v%5E2%2Bhighlight%282v%2B7v%29%2B14 Replace the second term 9v with 2v%2B7v.


%28v%5E2%2B2v%29%2B%287v%2B14%29 Group the terms into two pairs.


v%28v%2B2%29%2B%287v%2B14%29 Factor out the GCF v from the first group.


v%28v%2B2%29%2B7%28v%2B2%29 Factor out 7 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.


%28v%2B7%29%28v%2B2%29 Combine like terms. Or factor out the common term v%2B2


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Answer:


So v%5E2%2B9v%2B14 factors to %28v%2B7%29%28v%2B2%29.


In other words, v%5E2%2B9v%2B14=%28v%2B7%29%28v%2B2%29.


Note: you can check the answer by expanding %28v%2B7%29%28v%2B2%29 to get v%5E2%2B9v%2B14 or by graphing the original expression and the answer (the two graphs should be identical).

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# 3



Looking at the expression r%5E2-2r-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 -2r with 6r-8r. Remember, 6 and -8 add to -2. So this shows us that 6r-8r=-2r.


r%5E2%2Bhighlight%286r-8r%29-48 Replace the second term -2r with 6r-8r.


%28r%5E2%2B6r%29%2B%28-8r-48%29 Group the terms into two pairs.


r%28r%2B6%29%2B%28-8r-48%29 Factor out the GCF r from the first group.


r%28r%2B6%29-8%28r%2B6%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.


%28r-8%29%28r%2B6%29 Combine like terms. Or factor out the common term r%2B6


===============================================================


Answer:


So r%5E2-2r-48 factors to %28r-8%29%28r%2B6%29.


In other words, r%5E2-2r-48=%28r-8%29%28r%2B6%29.


Note: you can check the answer by expanding %28r-8%29%28r%2B6%29 to get r%5E2-2r-48 or by graphing the original expression and the answer (the two graphs should be identical).