# SOLUTION: factor the algebraic expressions completely -18a^2 + 17a +15

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Question 476853: factor the algebraic expressions completely
-18a^2 + 17a +15

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Factor out the GCF .

Now let's try to factor the inner expression

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Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last term is .

Now multiply the first coefficient by the last term to get .

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

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

Factors of :
1,2,3,5,6,9,10,15,18,27,30,45,54,90,135,270
-1,-2,-3,-5,-6,-9,-10,-15,-18,-27,-30,-45,-54,-90,-135,-270

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

These factors pair up and multiply to .
1*(-270) = -270
2*(-135) = -270
3*(-90) = -270
5*(-54) = -270
6*(-45) = -270
9*(-30) = -270
10*(-27) = -270
15*(-18) = -270
(-1)*(270) = -270
(-2)*(135) = -270
(-3)*(90) = -270
(-5)*(54) = -270
(-6)*(45) = -270
(-9)*(30) = -270
(-10)*(27) = -270
(-15)*(18) = -270

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

First NumberSecond NumberSum
1-2701+(-270)=-269
2-1352+(-135)=-133
3-903+(-90)=-87
5-545+(-54)=-49
6-456+(-45)=-39
9-309+(-30)=-21
10-2710+(-27)=-17
15-1815+(-18)=-3
-1270-1+270=269
-2135-2+135=133
-390-3+90=87
-554-5+54=49
-645-6+45=39
-930-9+30=21
-1027-10+27=17
-1518-15+18=3

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

So the two numbers and both multiply to and add to

Now replace the middle term with . Remember, and add to . So this shows us that .

Replace the second term with .

Group the terms into two pairs.

Factor out the GCF from the first group.

Factor out 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.

Combine like terms. Or factor out the common term

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So then factors further to

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