document.write( "Question 857602: Okay so I'm doing the factoring of trinomials when you have a coeffcient and your x is negative.
\n" ); document.write( "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? \n" ); document.write( "

Algebra.Com's Answer #516667 by richwmiller(17219) $\"\"$ $\"About$

You can put this solution on YOUR website!
You can also factor into ((-x-4)*(x-5)) or ((x+4)*(5-x))
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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 $\"x%5E2-x-20\"$

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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 $\"-1\"$ :

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First Number	Second Number	Sum
1	-20	1+(-20)=-19
2	-10	2+(-10)=-8
4	-5	4+(-5)=-1
-1	20	-1+20=19
-2	10	-2+10=8
-4	5	-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 $\"x%2B4\"$

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

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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).

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