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
Answer by jim_thompson5910(28476) About Me  (Show Source):
You can put this solution on YOUR website!

-18a%5E2%2B17a%2B15 Start with the given expression.


-%2818a%5E2-17a-15%29 Factor out the GCF -1.


Now let's try to factor the inner expression 18a%5E2-17a-15


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Looking at the expression 18a%5E2-17a-15, we can see that the first coefficient is 18, the second coefficient is -17, and the last term is -15.


Now multiply the first coefficient 18 by the last term -15 to get %2818%29%28-15%29=-270.


Now the question is: what two whole numbers multiply to -270 (the previous product) and add to the second coefficient -17?


To find these two numbers, we need to list all of the factors of -270 (the previous product).


Factors of -270:
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 -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
(-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 -17:


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 10 and -27 add to -17 (the middle coefficient).


So the two numbers 10 and -27 both multiply to -270 and add to -17


Now replace the middle term -17a with 10a-27a. Remember, 10 and -27 add to -17. So this shows us that 10a-27a=-17a.


18a%5E2%2Bhighlight%2810a-27a%29-15 Replace the second term -17a with 10a-27a.


%2818a%5E2%2B10a%29%2B%28-27a-15%29 Group the terms into two pairs.


2a%289a%2B5%29%2B%28-27a-15%29 Factor out the GCF 2a from the first group.


2a%289a%2B5%29-3%289a%2B5%29 Factor out 3 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.


%282a-3%29%289a%2B5%29 Combine like terms. Or factor out the common term 9a%2B5


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So -1%2818a%5E2-17a-15%29 then factors further to -%282a-3%29%289a%2B5%29


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Answer:


So -18a%5E2%2B17a%2B15 completely factors to -%282a-3%29%289a%2B5%29.


In other words, -18a%5E2%2B17a%2B15=-%282a-3%29%289a%2B5%29.


Note: you can check the answer by expanding -%282a-3%29%289a%2B5%29 to get -18a%5E2%2B17a%2B15 or by graphing the original expression and the answer (the two graphs should be identical).