SOLUTION: Factor the following trinomial, if possible. If the coefficient of the first term is negative, factor out -1 to make the first term positive. 10a^2 - 9a + 2

Algebra ->  Expressions -> SOLUTION: Factor the following trinomial, if possible. If the coefficient of the first term is negative, factor out -1 to make the first term positive. 10a^2 - 9a + 2      Log On


   

Question 317059: Factor the following trinomial, if possible. If the coefficient of the first term is negative, factor out -1 to make the first term positive.
10a^2 - 9a + 2
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 10a%5E2-9a%2B2, we can see that the first coefficient is 10, the second coefficient is -9, and the last term is 2.


Now multiply the first coefficient 10 by the last term 2 to get %2810%29%282%29=20.


Now the question is: what two whole numbers multiply to 20 (the previous product) and add to the second coefficient -9?


To find these two numbers, we need to list all of the factors of 20 (the previous product).


Factors of 20:
1,2,4,5,10,20
-1,-2,-4,-5,-10,-20


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


These factors pair up and multiply to 20.
1*20 = 20
2*10 = 20
4*5 = 20
(-1)*(-20) = 20
(-2)*(-10) = 20
(-4)*(-5) = 20

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -9:


First NumberSecond NumberSum
1201+20=21
2102+10=12
454+5=9
-1-20-1+(-20)=-21
-2-10-2+(-10)=-12
-4-5-4+(-5)=-9



From the table, we can see that the two numbers -4 and -5 add to -9 (the middle coefficient).


So the two numbers -4 and -5 both multiply to 20 and add to -9


Now replace the middle term -9a with -4a-5a. Remember, -4 and -5 add to -9. So this shows us that -4a-5a=-9a.


10a%5E2%2Bhighlight%28-4a-5a%29%2B2 Replace the second term -9a with -4a-5a.


%2810a%5E2-4a%29%2B%28-5a%2B2%29 Group the terms into two pairs.


2a%285a-2%29%2B%28-5a%2B2%29 Factor out the GCF 2a from the first group.


2a%285a-2%29-1%285a-2%29 Factor out 1 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-1%29%285a-2%29 Combine like terms. Or factor out the common term 5a-2


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Answer:


So 10a%5E2-9a%2B2 factors to %282a-1%29%285a-2%29.


In other words, 10a%5E2-9a%2B2=%282a-1%29%285a-2%29.


Note: you can check the answer by expanding %282a-1%29%285a-2%29 to get 10a%5E2-9a%2B2 or by graphing the original expression and the answer (the two graphs should be identical).