document.write( "Question 186892: 2xsquared-12xy-32ysquared \n" ); document.write( "

Algebra.Com's Answer #140091 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
I'm assuming that you want to factor.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"2x%5E2-12xy-32y%5E2\"$ Start with the given expression\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"2%28x%5E2-6xy-16y%5E2%29\"$ Factor out the GCF $\"2\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"x%5E2-6xy-16y%5E2\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "------------------------------------------------------------\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Looking at $\"x%5E2-6xy-16y%5E2\"$ we can see that the first term is $\"x%5E2\"$ and the last term is $\"-16y%5E2\"$ where the coefficients are 1 and -16 respectively.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 1 and the last coefficient -16 to get -16. Now what two numbers multiply to -16 and add to the middle coefficient -6? Let's list all of the factors of -16:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of -16:\r
\n" ); document.write( "\n" ); document.write( "1,2,4,8\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-8 ...List the negative factors as well. This will allow us to find all possible combinations\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to -16\r
\n" ); document.write( "\n" ); document.write( "(1)*(-16)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-8)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(16)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(8)\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "note: remember, the product of a negative and a positive number is a negative number\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now which of these pairs add to -6? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -6\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "

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

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From this list we can see that 2 and -8 add up to -6 and multiply to -16\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"x%5E2-6xy-16y%5E2\"$ , replace $\"-6xy\"$ with $\"2xy-8xy\"$ (notice $\"2xy-8xy\"$ combines to $\"-6xy\"$ . So it is equivalent to $\"-6xy\"$ )\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%282xy-8xy%29%2B-16y%5E2\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's factor $\"1x%5E2%2B2xy-8xy-16y%5E2\"$ by grouping:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2%2B2xy%29%2B%28-8xy-16y%5E2%29\"$ Group like terms\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B2y%29-8y%28x%2B2y%29\"$ Factor out the GCF of $\"x\"$ out of the first group. Factor out the GCF of $\"-8y\"$ out of the second group\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28x-8y%29%28x%2B2y%29\"$ Since we have a common term of $\"x%2B2y\"$ , we can combine like terms\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So $\"x%5E2%2B2xy-8xy-16y%5E2\"$ factors to $\"%28x-8y%29%28x%2B2y%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So this also means that $\"x%5E2-6xy-16y%5E2\"$ factors to $\"%28x-8y%29%28x%2B2y%29\"$ (since $\"x%5E2-6xy-16y%5E2\"$ is equivalent to $\"x%5E2%2B2xy-8xy-16y%5E2\"$ )\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "------------------------------------------------------------\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So our expression goes from $\"2%28x%5E2-6xy-16y%5E2%29\"$ and factors further to $\"2%28x-8y%29%28x%2B2y%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "------------------\r
\n" ); document.write( "\n" ); document.write( "Answer:\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So $\"2x%5E2-12xy-32y%5E2\"$ factors to $\"2%28x-8y%29%28x%2B2y%29\"$ \r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );