document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=143549>Question 143549</A>: <I><SPAN STYLE=\"word-break: break-all;\"> x^2-2xy-15y^2<BR>\n" );
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document.write( "5x^2+25x+30<BR>\n" );
document.write( "x^2-xy-12y^2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #104468 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=104468\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Do you want to factor?\r<BR>\n" );
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document.write( "I'll do the first two to get you started\r<BR>\n" );
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document.write( "# 1\r<BR>\n" );
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document.write( "Looking at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2-2xy-15y%5E2 ALIGN=MIDDLE  ALT=\"1x%5E2-2xy-15y%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=1x%5E2 ALIGN=MIDDLE  ALT=\"1x%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=-15y%5E2 ALIGN=MIDDLE  ALT=\"-15y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > where the coefficients are 1 and -15 respectively.\r<BR>\n" );
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document.write( "Now multiply the first coefficient 1 and the last coefficient -15 to get -15. Now what two numbers multiply to -15 and add to the  middle coefficient -2? Let's list all of the factors of -15:\r<BR>\n" );
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document.write( "Factors of -15:\r<BR>\n" );
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document.write( "1,3,5,15\r<BR>\n" );
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document.write( "-1,-3,-5,-15 ...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 -15\r<BR>\n" );
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document.write( "(1)*(-15)\r<BR>\n" );
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document.write( "(3)*(-5)\r<BR>\n" );
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document.write( "(-1)*(15)\r<BR>\n" );
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document.write( "(-3)*(5)\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 -2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -2\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\">-15</TD><TD >1+(-15)=-14</TD></tr><tr><TD  align=\"center\">3</TD><TD  align=\"center\">-5</TD><TD >3+(-5)=-2</TD></tr><tr><TD  align=\"center\">-1</TD><TD  align=\"center\">15</TD><TD >-1+15=14</TD></tr><tr><TD  align=\"center\">-3</TD><TD  align=\"center\">5</TD><TD >-3+5=2</TD></tr>\n" );
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document.write( "From this list we can see that 3 and -5 add up to -2 and multiply to -15\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=1x%5E2-2xy-15y%5E2 ALIGN=MIDDLE  ALT=\"1x%5E2-2xy-15y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2xy ALIGN=MIDDLE  ALT=\"-2xy\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3xy%2B-5xy ALIGN=MIDDLE  ALT=\"3xy%2B-5xy\"   DISABLED_style= \"max-width:90%; height:auto;\" > (notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3xy%2B-5xy ALIGN=MIDDLE  ALT=\"3xy%2B-5xy\"   DISABLED_style= \"max-width:90%; height:auto;\" > adds up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2xy ALIGN=MIDDLE  ALT=\"-2xy\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So it is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2xy ALIGN=MIDDLE  ALT=\"-2xy\"   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=1x%5E2%2Bhighlight%283xy%2B-5xy%29%2B-15y%5E2 ALIGN=MIDDLE  ALT=\"1x%5E2%2Bhighlight%283xy%2B-5xy%29%2B-15y%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=1x%5E2%2B3xy-5xy-15y%5E2 ALIGN=MIDDLE  ALT=\"1x%5E2%2B3xy-5xy-15y%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=%281x%5E2%2B3xy%29%2B%28-5xy-15y%5E2%29 ALIGN=MIDDLE  ALT=\"%281x%5E2%2B3xy%29%2B%28-5xy-15y%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%2B3y%29-5y%28x%2B3y%29 ALIGN=MIDDLE  ALT=\"x%28x%2B3y%29-5y%28x%2B3y%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=-5y ALIGN=MIDDLE  ALT=\"-5y\"   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-5y%29%28x%2B3y%29 ALIGN=MIDDLE  ALT=\"%28x-5y%29%28x%2B3y%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%2B3y ALIGN=MIDDLE  ALT=\"x%2B3y\"   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=1x%5E2%2B3xy-5xy-15y%5E2 ALIGN=MIDDLE  ALT=\"1x%5E2%2B3xy-5xy-15y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-5y%29%28x%2B3y%29 ALIGN=MIDDLE  ALT=\"%28x-5y%29%28x%2B3y%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=1x%5E2-2xy-15y%5E2 ALIGN=MIDDLE  ALT=\"1x%5E2-2xy-15y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-5y%29%28x%2B3y%29 ALIGN=MIDDLE  ALT=\"%28x-5y%29%28x%2B3y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2-2xy-15y%5E2 ALIGN=MIDDLE  ALT=\"1x%5E2-2xy-15y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2%2B3xy-5xy-15y%5E2 ALIGN=MIDDLE  ALT=\"1x%5E2%2B3xy-5xy-15y%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%5E2-2xy-15y%5E2 ALIGN=MIDDLE  ALT=\"x%5E2-2xy-15y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-5y%29%28x%2B3y%29 ALIGN=MIDDLE  ALT=\"%28x-5y%29%28x%2B3y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "# 2\r<BR>\n" );
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document.write( "Looking at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Bx%2B1 ALIGN=MIDDLE  ALT=\"x%5E2%2Bx%2B1\"   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=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > where the coefficients are 1 and 1 respectively.\r<BR>\n" );
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document.write( "Now multiply the first coefficient 1 and the last coefficient 1 to get 1. Now what two numbers multiply to 1 and add to the  middle coefficient 1? Let's list all of the factors of 1:\r<BR>\n" );
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document.write( "Factors of 1:\r<BR>\n" );
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document.write( "1\r<BR>\n" );
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document.write( "-1 ...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 1\r<BR>\n" );
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document.write( "1*1\r<BR>\n" );
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document.write( "(-1)*(-1)\r<BR>\n" );
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document.write( "note: remember two negative numbers multiplied together make a positive number\r<BR>\n" );
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document.write( "Now which of these pairs add to 1? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 1\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\">1</TD><TD >1+1=2</TD></tr><tr><TD  align=\"center\">-1</TD><TD  align=\"center\">-1</TD><TD >-1+(-1)=-2</TD></tr>\n" );
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document.write( "None of these pairs of factors add to 1. So the expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Bx%2B1 ALIGN=MIDDLE  ALT=\"x%5E2%2Bx%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > cannot be factored<BR>\n" );
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