document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=368662>Question 368662</A>: <I><SPAN STYLE=\"word-break: break-all;\"> please help me factor this trinomial 6p^2+5p-6 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #262689 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=262689\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "Looking at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6p%5E2%2B5p-6 ALIGN=MIDDLE  ALT=\"6p%5E2%2B5p-6\"   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=6p%5E2 ALIGN=MIDDLE  ALT=\"6p%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=-6 ALIGN=MIDDLE  ALT=\"-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > where the coefficients are 6 and -6 respectively.\r<BR>\n" );
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document.write( "Now multiply the first coefficient 6 and the last coefficient -6 to get -36. Now what two numbers multiply to -36 and add to the  middle coefficient 5? Let's list all of the factors of -36:\r<BR>\n" );
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document.write( "Factors of -36:\r<BR>\n" );
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document.write( "1,2,3,4,6,9,12,18\r<BR>\n" );
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document.write( "-1,-2,-3,-4,-6,-9,-12,-18 ...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 -36\r<BR>\n" );
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document.write( "(1)*(-36)\r<BR>\n" );
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document.write( "(2)*(-18)\r<BR>\n" );
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document.write( "(3)*(-12)\r<BR>\n" );
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document.write( "(4)*(-9)\r<BR>\n" );
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document.write( "(-1)*(36)\r<BR>\n" );
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document.write( "(-2)*(18)\r<BR>\n" );
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document.write( "(-3)*(12)\r<BR>\n" );
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document.write( "(-4)*(9)\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 5? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 5\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\">-36</TD><TD >1+(-36)=-35</TD></tr><tr><TD  align=\"center\">2</TD><TD  align=\"center\">-18</TD><TD >2+(-18)=-16</TD></tr><tr><TD  align=\"center\">3</TD><TD  align=\"center\">-12</TD><TD >3+(-12)=-9</TD></tr><tr><TD  align=\"center\">4</TD><TD  align=\"center\">-9</TD><TD >4+(-9)=-5</TD></tr><tr><TD  align=\"center\">-1</TD><TD  align=\"center\">36</TD><TD >-1+36=35</TD></tr><tr><TD  align=\"center\">-2</TD><TD  align=\"center\">18</TD><TD >-2+18=16</TD></tr><tr><TD  align=\"center\">-3</TD><TD  align=\"center\">12</TD><TD >-3+12=9</TD></tr><tr><TD  align=\"center\">-4</TD><TD  align=\"center\">9</TD><TD >-4+9=5</TD></tr>\n" );
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document.write( "From this list we can see that -4 and 9 add up to 5 and multiply to -36\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=6p%5E2%2B5p-6 ALIGN=MIDDLE  ALT=\"6p%5E2%2B5p-6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5p ALIGN=MIDDLE  ALT=\"5p\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4p%2B9p ALIGN=MIDDLE  ALT=\"-4p%2B9p\"   DISABLED_style= \"max-width:90%; height:auto;\" > (notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4p%2B9p ALIGN=MIDDLE  ALT=\"-4p%2B9p\"   DISABLED_style= \"max-width:90%; height:auto;\" > adds up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5p ALIGN=MIDDLE  ALT=\"5p\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So it is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5p ALIGN=MIDDLE  ALT=\"5p\"   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=6p%5E2%2Bhighlight%28-4p%2B9p%29%2B-6 ALIGN=MIDDLE  ALT=\"6p%5E2%2Bhighlight%28-4p%2B9p%29%2B-6\"   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=6p%5E2-4p%2B9p-6 ALIGN=MIDDLE  ALT=\"6p%5E2-4p%2B9p-6\"   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=%286p%5E2-4p%29%2B%289p-6%29 ALIGN=MIDDLE  ALT=\"%286p%5E2-4p%29%2B%289p-6%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=2p%283p-2%29%2B3%283p-2%29 ALIGN=MIDDLE  ALT=\"2p%283p-2%29%2B3%283p-2%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=2p ALIGN=MIDDLE  ALT=\"2p\"   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=3 ALIGN=MIDDLE  ALT=\"3\"   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=%282p%2B3%29%283p-2%29 ALIGN=MIDDLE  ALT=\"%282p%2B3%29%283p-2%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=3p-2 ALIGN=MIDDLE  ALT=\"3p-2\"   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=6p%5E2-4p%2B9p-6 ALIGN=MIDDLE  ALT=\"6p%5E2-4p%2B9p-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282p%2B3%29%283p-2%29 ALIGN=MIDDLE  ALT=\"%282p%2B3%29%283p-2%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=6p%5E2%2B5p-6 ALIGN=MIDDLE  ALT=\"6p%5E2%2B5p-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282p%2B3%29%283p-2%29 ALIGN=MIDDLE  ALT=\"%282p%2B3%29%283p-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6p%5E2%2B5p-6 ALIGN=MIDDLE  ALT=\"6p%5E2%2B5p-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6p%5E2-4p%2B9p-6 ALIGN=MIDDLE  ALT=\"6p%5E2-4p%2B9p-6\"   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=6p%5E2%2B5p-6 ALIGN=MIDDLE  ALT=\"6p%5E2%2B5p-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282p%2B3%29%283p-2%29 ALIGN=MIDDLE  ALT=\"%282p%2B3%29%283p-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
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