SOLUTION: Hello, I need a little help please. I need to Factor Completely the following problem. 4x^2 + 20x - 96 Any help that you can give would be greatly appreciated. Thank you

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Hello, I need a little help please. I need to Factor Completely the following problem. 4x^2 + 20x - 96 Any help that you can give would be greatly appreciated. Thank you       Log On


   

Question 201377: Hello,
I need a little help please. I need to Factor Completely the following problem.
4x^2 + 20x - 96
Any help that you can give would be greatly appreciated.
Thank you in advance for your help.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

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4x%5E2%2B20x-96 Start with the given expression.


4%28x%5E2%2B5x-24%29 Factor out the GCF 4.


Now let's try to factor the inner expression x%5E2%2B5x-24


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Looking at the expression x%5E2%2B5x-24, we can see that the first coefficient is 1, the second coefficient is 5, and the last term is -24.


Now multiply the first coefficient 1 by the last term -24 to get %281%29%28-24%29=-24.


Now the question is: what two whole numbers multiply to -24 (the previous product) and add to the second coefficient 5?


To find these two numbers, we need to list all of the factors of -24 (the previous product).


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


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


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

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 5:


First NumberSecond NumberSum
1-241+(-24)=-23
2-122+(-12)=-10
3-83+(-8)=-5
4-64+(-6)=-2
-124-1+24=23
-212-2+12=10
-38-3+8=5
-46-4+6=2



From the table, we can see that the two numbers -3 and 8 add to 5 (the middle coefficient).


So the two numbers -3 and 8 both multiply to -24 and add to 5


Now replace the middle term 5x with -3x%2B8x. Remember, -3 and 8 add to 5. So this shows us that -3x%2B8x=5x.


x%5E2%2Bhighlight%28-3x%2B8x%29-24 Replace the second term 5x with -3x%2B8x.


%28x%5E2-3x%29%2B%288x-24%29 Group the terms into two pairs.


x%28x-3%29%2B%288x-24%29 Factor out the GCF x from the first group.


x%28x-3%29%2B8%28x-3%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-3%29 Combine like terms. Or factor out the common term x-3


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So 4%28x%5E2%2B5x-24%29 then factors further to 4%28x%2B8%29%28x-3%29


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Answer:


So 4x%5E2%2B20x-96 completely factors to 4%28x%2B8%29%28x-3%29.


Note: you can check the answer by expanding 4%28x%2B8%29%28x-3%29 to get 4x%5E2%2B20x-96 or by graphing the original expression and the answer (the two graphs should be identical).

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