document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=251213>Question 251213</A>: <I><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+6xy+-+40y%5E2 ALIGN=MIDDLE  ALT=\"x%5E2+-+6xy+-+40y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Factor completely. \r<BR>\n" );
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document.write( "I do not understand how to do this. Its frustrating because I think I understand but I am still getting the answer wrong. Please help! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #182943 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=182943\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "Looking at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-6xy-40y%5E2 ALIGN=MIDDLE  ALT=\"x%5E2-6xy-40y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > we can see that the first term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the last term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-40y%5E2 ALIGN=MIDDLE  ALT=\"-40y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > where the coefficients are 1 and -40 respectively.\r<BR>\n" );
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document.write( "Now multiply the first coefficient 1 and the last coefficient -40 to get -40. Now what two numbers multiply to -40 and add to the  middle coefficient -6? Let's list all of the factors of -40:\r<BR>\n" );
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document.write( "Factors of -40:\r<BR>\n" );
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document.write( "1,2,4,5,8,10,20,40\r<BR>\n" );
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document.write( "-1,-2,-4,-5,-8,-10,-20,-40 ...List the negative factors as well. This will allow us to find all possible combinations\r<BR>\n" );
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document.write( "These factors pair up and multiply to -40\r<BR>\n" );
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document.write( "(1)*(-40)\r<BR>\n" );
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document.write( "(2)*(-20)\r<BR>\n" );
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document.write( "(4)*(-10)\r<BR>\n" );
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document.write( "(5)*(-8)\r<BR>\n" );
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document.write( "(-1)*(40)\r<BR>\n" );
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document.write( "(-2)*(20)\r<BR>\n" );
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document.write( "(-4)*(10)\r<BR>\n" );
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document.write( "(-5)*(8)\r<BR>\n" );
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document.write( "note: remember, the product of a negative and a positive number is a negative number\r<BR>\n" );
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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<BR>\n" );
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document.write( "<th>First Number</th><th>Second Number</th><th>Sum</th><tr><TD  align=\"center\">1</TD><TD  align=\"center\">-40</TD><TD >1+(-40)=-39</TD></tr><tr><TD  align=\"center\">2</TD><TD  align=\"center\">-20</TD><TD >2+(-20)=-18</TD></tr><tr><TD  align=\"center\">4</TD><TD  align=\"center\">-10</TD><TD >4+(-10)=-6</TD></tr><tr><TD  align=\"center\">5</TD><TD  align=\"center\">-8</TD><TD >5+(-8)=-3</TD></tr><tr><TD  align=\"center\">-1</TD><TD  align=\"center\">40</TD><TD >-1+40=39</TD></tr><tr><TD  align=\"center\">-2</TD><TD  align=\"center\">20</TD><TD >-2+20=18</TD></tr><tr><TD  align=\"center\">-4</TD><TD  align=\"center\">10</TD><TD >-4+10=6</TD></tr><tr><TD  align=\"center\">-5</TD><TD  align=\"center\">8</TD><TD >-5+8=3</TD></tr>\n" );
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document.write( "From this list we can see that 4 and -10 add up to -6 and multiply to -40\r<BR>\n" );
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document.write( "Now looking at the expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-6xy-40y%5E2 ALIGN=MIDDLE  ALT=\"x%5E2-6xy-40y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6xy ALIGN=MIDDLE  ALT=\"-6xy\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4xy-10xy ALIGN=MIDDLE  ALT=\"4xy-10xy\"   DISABLED_style= \"max-width:90%; height:auto;\" > (notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4xy-10xy ALIGN=MIDDLE  ALT=\"4xy-10xy\"   DISABLED_style= \"max-width:90%; height:auto;\" > combines back to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6xy ALIGN=MIDDLE  ALT=\"-6xy\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So it is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6xy ALIGN=MIDDLE  ALT=\"-6xy\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Bhighlight%284xy-10xy%29-40y%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2Bhighlight%284xy-10xy%29-40y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now let's factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B4xy-10xy-40y%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2B4xy-10xy-40y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > by grouping:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2B4xy%29%2B%28-10xy-40y%5E2%29 ALIGN=MIDDLE  ALT=\"%28x%5E2%2B4xy%29%2B%28-10xy-40y%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Group like terms\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%28x%2B4y%29-10y%28x%2B4y%29 ALIGN=MIDDLE  ALT=\"x%28x%2B4y%29-10y%28x%2B4y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out the GCF of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > out of the first group. Factor out the GCF of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-10y ALIGN=MIDDLE  ALT=\"-10y\"   DISABLED_style= \"max-width:90%; height:auto;\" > out of the second group\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-10y%29%28x%2B4y%29 ALIGN=MIDDLE  ALT=\"%28x-10y%29%28x%2B4y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Since we have a common term of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B4y ALIGN=MIDDLE  ALT=\"x%2B4y\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can combine like terms\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B4xy-10xy-40y%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2B4xy-10xy-40y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-10y%29%28x%2B4y%29 ALIGN=MIDDLE  ALT=\"%28x-10y%29%28x%2B4y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So this also means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-6xy-40y%5E2 ALIGN=MIDDLE  ALT=\"x%5E2-6xy-40y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-10y%29%28x%2B4y%29 ALIGN=MIDDLE  ALT=\"%28x-10y%29%28x%2B4y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-6xy-40y%5E2 ALIGN=MIDDLE  ALT=\"x%5E2-6xy-40y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B4xy-10xy-40y%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2B4xy-10xy-40y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "     Answer:\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-6xy-40y%5E2 ALIGN=MIDDLE  ALT=\"x%5E2-6xy-40y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-10y%29%28x%2B4y%29 ALIGN=MIDDLE  ALT=\"%28x-10y%29%28x%2B4y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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