SOLUTION: Factor by grouping. (If the expression is nonfactorable enter PRIME.) 50x2 – 50x – 12

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Question 729201: Factor by grouping. (If the expression is nonfactorable enter PRIME.)
50x2 – 50x – 12
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

50x%5E2-50x-12 Start with the given expression.


2%2825x%5E2-25x-6%29 Factor out the GCF 2.


Now let's try to factor the inner expression 25x%5E2-25x-6


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Looking at the expression 25x%5E2-25x-6, we can see that the first coefficient is 25, the second coefficient is -25, and the last term is -6.


Now multiply the first coefficient 25 by the last term -6 to get %2825%29%28-6%29=-150.


Now the question is: what two whole numbers multiply to -150 (the previous product) and add to the second coefficient -25?


To find these two numbers, we need to list all of the factors of -150 (the previous product).


Factors of -150:
1,2,3,5,6,10,15,25,30,50,75,150
-1,-2,-3,-5,-6,-10,-15,-25,-30,-50,-75,-150


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


These factors pair up and multiply to -150.
1*(-150) = -150
2*(-75) = -150
3*(-50) = -150
5*(-30) = -150
6*(-25) = -150
10*(-15) = -150
(-1)*(150) = -150
(-2)*(75) = -150
(-3)*(50) = -150
(-5)*(30) = -150
(-6)*(25) = -150
(-10)*(15) = -150

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -25:


First NumberSecond NumberSum
1-1501+(-150)=-149
2-752+(-75)=-73
3-503+(-50)=-47
5-305+(-30)=-25
6-256+(-25)=-19
10-1510+(-15)=-5
-1150-1+150=149
-275-2+75=73
-350-3+50=47
-530-5+30=25
-625-6+25=19
-1015-10+15=5



From the table, we can see that the two numbers 5 and -30 add to -25 (the middle coefficient).


So the two numbers 5 and -30 both multiply to -150 and add to -25


Now replace the middle term -25x with 5x-30x. Remember, 5 and -30 add to -25. So this shows us that 5x-30x=-25x.


25x%5E2%2Bhighlight%285x-30x%29-6 Replace the second term -25x with 5x-30x.


%2825x%5E2%2B5x%29%2B%28-30x-6%29 Group the terms into two pairs.


5x%285x%2B1%29%2B%28-30x-6%29 Factor out the GCF 5x from the first group.


5x%285x%2B1%29-6%285x%2B1%29 Factor out 6 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.


%285x-6%29%285x%2B1%29 Combine like terms. Or factor out the common term 5x%2B1


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So 2%2825x%5E2-25x-6%29 then factors further to 2%285x-6%29%285x%2B1%29


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Answer:


So 50x%5E2-50x-12 completely factors to 2%285x-6%29%285x%2B1%29.


In other words, 50x%5E2-50x-12=2%285x-6%29%285x%2B1%29.


Note: you can check the answer by expanding 2%285x-6%29%285x%2B1%29 to get 50x%5E2-50x-12 or by graphing the original expression and the answer (the two graphs should be identical).