document.write( "Question 896901: Are there two integers with a product of negative 16 and a sum of negative 4? \n" ); document.write( "

Algebra.Com's Answer #543847 by richwmiller(17219) $\"\"$ $\"About$

You can put this solution on YOUR website!
x*y=-16,
\n" ); document.write( "x+y=-4
\n" ); document.write( "y=-x-4
\n" ); document.write( "x*(-x-4)=-16\r
\n" ); document.write( "\n" ); document.write( "-x^2-4x+16=0
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

$\"-x%5E2-4%2Ax%2B16\"$ Start with the given expression.

$\"-%28x%5E2%2B4x-16%29\"$ Factor out the GCF $\"-1\"$ .

Now let's try to factor the inner expression $\"x%5E2%2B4x-16\"$

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

Looking at the expression $\"x%5E2%2B4x-16\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"4\"$ , and the last term is $\"-16\"$ .

Now multiply the first coefficient $\"1\"$ by the last term $\"-16\"$ to get $\"%281%29%28-16%29=-16\"$ .

Now the question is: what two whole numbers multiply to $\"-16\"$ (the previous product) and add to the second coefficient $\"4\"$ ?

To find these two numbers, we need to list all of the factors of $\"-16\"$ (the previous product).

Factors of $\"-16\"$ :

1,2,4,8,16

-1,-2,-4,-8,-16

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

These factors pair up and multiply to $\"-16\"$ .

1*(-16) = -16
2*(-8) = -16
4*(-4) = -16
(-1)*(16) = -16
(-2)*(8) = -16
(-4)*(4) = -16

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"4\"$ :

\n" ); document.write( "

First Number	Second Number	Sum
1	-16	1+(-16)=-15
2	-8	2+(-8)=-6
4	-4	4+(-4)=0
-1	16	-1+16=15
-2	8	-2+8=6
-4	4	-4+4=0

From the table, we can see that there are no pairs of numbers which add to $\"4\"$ . So $\"x%5E2%2B4x-16\"$ cannot be factored.

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

Answer:

So $\"-x%5E2-4%2Ax%2B16\"$ simply factors to $\"-%28x%5E2%2B4x-16%29\"$

In other words, $\"-x%5E2-4%2Ax%2B16=-%28x%5E2%2B4x-16%29\"$ .

\n" ); document.write( " \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );