SOLUTION: Factor completely 30x^2 + 4x – 2

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Question 189832: Factor completely
30x^2 + 4x – 2
Found 2 solutions by josmiceli, jim_thompson5910:
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
30x%5E2+%2B+4x+-+2
use quadratic formula
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
a+=+30
b+=+4
c+=+-2
x+=+%28-4+%2B-+sqrt%28+4%5E2-4%2A30%2A%28-2%29+%29%29%2F%282%2A30%29+
x+=+%28-4+%2B-+sqrt%28+16+%2B+240+%29%29%2F60+
x+=+%28-4+%2B-+sqrt%28+256+%29%29%2F60+
x+=+%28-4+%2B-+16%29%2F60+
x+=+%28-4+%2B+16%29%2F60
x+=+1%2F5
and
x+=+%28-4+-+16%29%2F60
x+=+-1%2F3
The factors are:
%28x+-+1%2F5%29%28x+%2B+1%2F3%29

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Looking at the expression 30x%5E2%2B4x-2, we can see that the first coefficient is 30, the second coefficient is 4, and the last term is -2.


Now multiply the first coefficient 30 by the last term -2 to get %2830%29%28-2%29=-60.


Now the question is: what two whole numbers multiply to -60 (the previous product) and add to the second coefficient 4?


To find these two numbers, we need to list all of the factors of -60 (the previous product).


Factors of -60:
1,2,3,4,5,6,10,12,15,20,30,60
-1,-2,-3,-4,-5,-6,-10,-12,-15,-20,-30,-60


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


These factors pair up and multiply to -60.
1*(-60)
2*(-30)
3*(-20)
4*(-15)
5*(-12)
6*(-10)
(-1)*(60)
(-2)*(30)
(-3)*(20)
(-4)*(15)
(-5)*(12)
(-6)*(10)

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 4:


First NumberSecond NumberSum
1-601+(-60)=-59
2-302+(-30)=-28
3-203+(-20)=-17
4-154+(-15)=-11
5-125+(-12)=-7
6-106+(-10)=-4
-160-1+60=59
-230-2+30=28
-320-3+20=17
-415-4+15=11
-512-5+12=7
-610-6+10=4



From the table, we can see that the two numbers -6 and 10 add to 4 (the middle coefficient).


So the two numbers -6 and 10 both multiply to -60 and add to 4


Now replace the middle term 4x with -6x%2B10x. Remember, -6 and 10 add to 4. So this shows us that -6x%2B10x=4x.


30x%5E2%2Bhighlight%28-6x%2B10x%29-2 Replace the second term 4x with -6x%2B10x.


%2830x%5E2-6x%29%2B%2810x-2%29 Group the terms into two pairs.


6x%285x-1%29%2B%2810x-2%29 Factor out the GCF 6x from the first group.


6x%285x-1%29%2B2%285x-1%29 Factor out 2 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.


%286x%2B2%29%285x-1%29 Combine like terms. Or factor out the common term 5x-1

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Answer:


So 30x%5E2%2B4x-2 factors to %286x%2B2%29%285x-1%29.


Note: you can check the answer by FOILing %286x%2B2%29%285x-1%29 to get 30x%5E2%2B4x-2 or by graphing the original expression and the answer (the two graphs should be identical).