SOLUTION: how can i factor this completely? 2x^2+7x-15

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Question 270245: how can i factor this completely?
2x^2+7x-15
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression 2x%5E2%2B7x-15, we can see that the first coefficient is 2, the second coefficient is 7, and the last term is -15.



Now multiply the first coefficient 2 by the last term -15 to get %282%29%28-15%29=-30.



Now the question is: what two whole numbers multiply to -30 (the previous product) and add to the second coefficient 7?



To find these two numbers, we need to list all of the factors of -30 (the previous product).



Factors of -30:

1,2,3,5,6,10,15,30

-1,-2,-3,-5,-6,-10,-15,-30



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



These factors pair up and multiply to -30.

1*(-30) = -30
2*(-15) = -30
3*(-10) = -30
5*(-6) = -30
(-1)*(30) = -30
(-2)*(15) = -30
(-3)*(10) = -30
(-5)*(6) = -30


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 7:



First NumberSecond NumberSum
1-301+(-30)=-29
2-152+(-15)=-13
3-103+(-10)=-7
5-65+(-6)=-1
-130-1+30=29
-215-2+15=13
-310-3+10=7
-56-5+6=1




From the table, we can see that the two numbers -3 and 10 add to 7 (the middle coefficient).



So the two numbers -3 and 10 both multiply to -30 and add to 7



Now replace the middle term 7x with -3x%2B10x. Remember, -3 and 10 add to 7. So this shows us that -3x%2B10x=7x.



2x%5E2%2Bhighlight%28-3x%2B10x%29-15 Replace the second term 7x with -3x%2B10x.



%282x%5E2-3x%29%2B%2810x-15%29 Group the terms into two pairs.



x%282x-3%29%2B%2810x-15%29 Factor out the GCF x from the first group.



x%282x-3%29%2B5%282x-3%29 Factor out 5 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%2B5%29%282x-3%29 Combine like terms. Or factor out the common term 2x-3



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Answer:



So 2%2Ax%5E2%2B7%2Ax-15 factors to %28x%2B5%29%282x-3%29.



In other words, 2%2Ax%5E2%2B7%2Ax-15=%28x%2B5%29%282x-3%29.



Note: you can check the answer by expanding %28x%2B5%29%282x-3%29 to get 2%2Ax%5E2%2B7%2Ax-15 or by graphing the original expression and the answer (the two graphs should be identical).