SOLUTION: Okay so I'm doing the factoring of trinomials when you have a coeffcient and your x is negative. The problem is -x^2+x+20 and you have to factor it out and find the roots, and I'm

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Okay so I'm doing the factoring of trinomials when you have a coeffcient and your x is negative. The problem is -x^2+x+20 and you have to factor it out and find the roots, and I'm      Log On


   

Question 857602: Okay so I'm doing the factoring of trinomials when you have a coeffcient and your x is negative.
The problem is -x^2+x+20 and you have to factor it out and find the roots, and I'm trying to factor it out so that when I distribute, I get the original equation, but I'm not getting it. As far as I know the roots are -4, and 5 but I'm not understanding why those are the roots. Can you help with the factoring and the roots?
Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
You can also factor into ((-x-4)*(x-5)) or ((x+4)*(5-x))
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


-x%5E2%2Bx%2B20 Start with the given expression.



-%28x%5E2-x-20%29 Factor out the GCF -1.



Now let's try to factor the inner expression x%5E2-x-20



---------------------------------------------------------------



Looking at the expression x%5E2-x-20, we can see that the first coefficient is 1, the second coefficient is -1, and the last term is -20.



Now multiply the first coefficient 1 by the last term -20 to get %281%29%28-20%29=-20.



Now the question is: what two whole numbers multiply to -20 (the previous product) and add to the second coefficient -1?



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 -1:



First NumberSecond NumberSum
1-201+(-20)=-19
2-102+(-10)=-8
4-54+(-5)=-1
-120-1+20=19
-210-2+10=8
-45-4+5=1




From the table, we can see that the two numbers 4 and -5 add to -1 (the middle coefficient).



So the two numbers 4 and -5 both multiply to -20 and add to -1



Now replace the middle term -1x with 4x-5x. Remember, 4 and -5 add to -1. So this shows us that 4x-5x=-1x.



x%5E2%2Bhighlight%284x-5x%29-20 Replace the second term -1x with 4x-5x.



%28x%5E2%2B4x%29%2B%28-5x-20%29 Group the terms into two pairs.



x%28x%2B4%29%2B%28-5x-20%29 Factor out the GCF x from the first group.



x%28x%2B4%29-5%28x%2B4%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-5%29%28x%2B4%29 Combine like terms. Or factor out the common term x%2B4



--------------------------------------------------



So -1%28x%5E2-x-20%29 then factors further to -%28x-5%29%28x%2B4%29



===============================================================



Answer:



So -x%5E2%2Bx%2B20 completely factors to -%28x-5%29%28x%2B4%29.



In other words, -x%5E2%2Bx%2B20=-%28x-5%29%28x%2B4%29.



Note: you can check the answer by expanding -%28x-5%29%28x%2B4%29 to get -x%5E2%2Bx%2B20 or by graphing the original expression and the answer (the two graphs should be identical).