document.write( "Question 116200: x^2-11xy-42y^2 \n" ); document.write( "

Algebra.Com's Answer #84508 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
\r
\n" ); document.write( "\n" ); document.write( "Looking at $\"x%5E2-11xy-42y%5E2\"$ we can see that the first term is $\"x%5E2\"$ and the last term is $\"-42y%5E2\"$ where the coefficients are 1 and -42 respectively.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 1 and the last coefficient -42 to get -42. Now what two numbers multiply to -42 and add to the middle coefficient -11? Let's list all of the factors of -42:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of -42:\r
\n" ); document.write( "\n" ); document.write( "1,2,3,6,7,14,21,42\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-6,-7,-14,-21,-42 ...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 -42\r
\n" ); document.write( "\n" ); document.write( "(1)*(-42)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-21)\r
\n" ); document.write( "\n" ); document.write( "(3)*(-14)\r
\n" ); document.write( "\n" ); document.write( "(6)*(-7)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(42)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(21)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(14)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(7)\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 -11? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -11\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "

First Number	Second Number	Sum
1	-42	1+(-42)=-41
2	-21	2+(-21)=-19
3	-14	3+(-14)=-11
6	-7	6+(-7)=-1
-1	42	-1+42=41
-2	21	-2+21=19
-3	14	-3+14=11
-6	7	-6+7=1

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From this list we can see that 3 and -14 add up to -11 and multiply to -42\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"x%5E2-11xy-42y%5E2\"$ , replace $\"-11xy\"$ with $\"3xy%2B-14xy\"$ (notice $\"3xy%2B-14xy\"$ adds up to $\"-11xy\"$ . So it is equivalent to $\"-11xy\"$ )\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%283xy%2B-14xy%29%2B-42y%5E2\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's factor $\"x%5E2%2B3xy-14xy-42y%5E2\"$ by grouping:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2%2B3xy%29%2B%28-14xy-42y%5E2%29\"$ Group like terms\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B3y%29-14y%28x%2B3y%29\"$ Factor out the GCF of $\"x\"$ out of the first group. Factor out the GCF of $\"-14y\"$ out of the second group\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28x-14y%29%28x%2B3y%29\"$ Since we have a common term of $\"x%2B3y\"$ , we can combine like terms\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So $\"x%5E2%2B3xy-14xy-42y%5E2\"$ factors to $\"%28x-14y%29%28x%2B3y%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So this also means that $\"x%5E2-11xy-42y%5E2\"$ factors to $\"%28x-14y%29%28x%2B3y%29\"$ (since $\"x%5E2-11xy-42y%5E2\"$ is equivalent to $\"x%5E2%2B3xy-14xy-42y%5E2\"$ )\r
\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 $\"x%5E2-11xy-42y%5E2\"$ factors to $\"%28x-14y%29%28x%2B3y%29\"$ \n" ); document.write( "

\n" );