SOLUTION: factor 3x^2 +13x -10

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Question 624249: factor 3x^2 +13x -10
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'm assuming you meant to write 3x^2+13x-10


Looking at the expression 3x%5E2%2B13x-10, we can see that the first coefficient is 3, the second coefficient is 13, and the last term is -10.


Now multiply the first coefficient 3 by the last term -10 to get %283%29%28-10%29=-30.


Now the question is: what two whole numbers multiply to -30 (the previous product) and add to the second coefficient 13?


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 13:


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 -2 and 15 add to 13 (the middle coefficient).


So the two numbers -2 and 15 both multiply to -30 and add to 13


Now replace the middle term 13x with -2x%2B15x. Remember, -2 and 15 add to 13. So this shows us that -2x%2B15x=13x.


3x%5E2%2Bhighlight%28-2x%2B15x%29-10 Replace the second term 13x with -2x%2B15x.


%283x%5E2-2x%29%2B%2815x-10%29 Group the terms into two pairs.


x%283x-2%29%2B%2815x-10%29 Factor out the GCF x from the first group.


x%283x-2%29%2B5%283x-2%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%283x-2%29 Combine like terms. Or factor out the common term 3x-2


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Answer:


So 3x%5E2%2B13x-10 factors to %28x%2B5%29%283x-2%29.


In other words, 3x%5E2%2B13x-10=%28x%2B5%29%283x-2%29.


Note: you can check the answer by expanding %28x%2B5%29%283x-2%29 to get 3x%5E2%2B13x-10 or by graphing the original expression and the answer (the two graphs should be identical).


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