document.write( "Question 219494: x² - 4xy – 12y² and a² - 25a factor completely (seperate problems)
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #165053 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
I'll do the first one to get you started.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Looking at $\"x%5E2-4xy-12y%5E2\"$ we can see that the first term is $\"x%5E2\"$ and the last term is $\"-12y%5E2\"$ where the coefficients are 1 and -12 respectively.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 1 and the last coefficient -12 to get -12. Now what two numbers multiply to -12 and add to the middle coefficient -4? Let's list all of the factors of -12:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of -12:\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,12\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-12 ...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 -12\r
\n" ); document.write( "\n" ); document.write( "(1)*(-12)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-6)\r
\n" ); document.write( "\n" ); document.write( "(3)*(-4)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(12)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(6)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(4)\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 -4? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -4\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "

First Number	Second Number	Sum
1	-12	1+(-12)=-11
2	-6	2+(-6)=-4
3	-4	3+(-4)=-1
-1	12	-1+12=11
-2	6	-2+6=4
-3	4	-3+4=1

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From this list we can see that 2 and -6 add up to -4 and multiply to -12\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"x%5E2-4xy-12y%5E2\"$ , replace $\"-4xy\"$ with $\"2xy-6xy\"$ (notice $\"2xy-6xy\"$ combines back to $\"-4xy\"$ . So it is equivalent to $\"-4xy\"$ )\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%282xy-6xy%29-12y%5E2\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's factor $\"x%5E2%2B2xy-6xy-12y%5E2\"$ by grouping:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2%2B2xy%29%2B%28-6xy-12y%5E2%29\"$ Group like terms\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B2y%29-6y%28x%2B2y%29\"$ Factor out the GCF of $\"x\"$ out of the first group. Factor out the GCF of $\"-6y\"$ out of the second group\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28x-6y%29%28x%2B2y%29\"$ Since we have a common term of $\"x%2B2y\"$ , we can combine like terms\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So $\"x%5E2%2B2xy-6xy-12y%5E2\"$ factors to $\"%28x-6y%29%28x%2B2y%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So this also means that $\"x%5E2-4xy-12y%5E2\"$ factors to $\"%28x-6y%29%28x%2B2y%29\"$ (since $\"x%5E2-4xy-12y%5E2\"$ is equivalent to $\"x%5E2%2B2xy-6xy-12y%5E2\"$ )\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( " Answer:\r
\n" ); document.write( "\n" ); document.write( "So $\"x%5E2-4xy-12y%5E2\"$ factors to $\"%28x-6y%29%28x%2B2y%29\"$
\n" ); document.write( " \n" ); document.write( "

\n" );