document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=115563>Question 115563</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Factor using AC Method.  9x^2-15x-6 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #84115 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=84115\">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=9x%5E2-15x-6 ALIGN=MIDDLE  ALT=\"9x%5E2-15x-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=9x%5E2 ALIGN=MIDDLE  ALT=\"9x%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 9 and -6 respectively.\r<BR>\n" );
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document.write( "Now multiply the first coefficient 9 and the last coefficient -6 to get -54. Now what two numbers multiply to -54 and add to the  middle coefficient -15? Let's list all of the factors of -54:\r<BR>\n" );
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document.write( "Factors of -54:\r<BR>\n" );
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document.write( "1,2,3,6,9,18,27,54\r<BR>\n" );
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document.write( "-1,-2,-3,-6,-9,-18,-27,-54 ...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 -54\r<BR>\n" );
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document.write( "(1)*(-54)\r<BR>\n" );
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document.write( "(2)*(-27)\r<BR>\n" );
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document.write( "(3)*(-18)\r<BR>\n" );
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document.write( "(6)*(-9)\r<BR>\n" );
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document.write( "(-1)*(54)\r<BR>\n" );
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document.write( "(-2)*(27)\r<BR>\n" );
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document.write( "(-3)*(18)\r<BR>\n" );
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document.write( "(-6)*(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 -15? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -15\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\">-54</TD><TD >1+(-54)=-53</TD></tr><tr><TD  align=\"center\">2</TD><TD  align=\"center\">-27</TD><TD >2+(-27)=-25</TD></tr><tr><TD  align=\"center\">3</TD><TD  align=\"center\">-18</TD><TD >3+(-18)=-15</TD></tr><tr><TD  align=\"center\">6</TD><TD  align=\"center\">-9</TD><TD >6+(-9)=-3</TD></tr><tr><TD  align=\"center\">-1</TD><TD  align=\"center\">54</TD><TD >-1+54=53</TD></tr><tr><TD  align=\"center\">-2</TD><TD  align=\"center\">27</TD><TD >-2+27=25</TD></tr><tr><TD  align=\"center\">-3</TD><TD  align=\"center\">18</TD><TD >-3+18=15</TD></tr><tr><TD  align=\"center\">-6</TD><TD  align=\"center\">9</TD><TD >-6+9=3</TD></tr>\n" );
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document.write( "From this list we can see that 3 and -18 add up to -15 and multiply to -54\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=9x%5E2-15x-6 ALIGN=MIDDLE  ALT=\"9x%5E2-15x-6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-15x ALIGN=MIDDLE  ALT=\"-15x\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B-18x ALIGN=MIDDLE  ALT=\"3x%2B-18x\"   DISABLED_style= \"max-width:90%; height:auto;\" > (notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B-18x ALIGN=MIDDLE  ALT=\"3x%2B-18x\"   DISABLED_style= \"max-width:90%; height:auto;\" > adds up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-15x ALIGN=MIDDLE  ALT=\"-15x\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So it is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-15x ALIGN=MIDDLE  ALT=\"-15x\"   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=9x%5E2%2Bhighlight%283x%2B-18x%29%2B-6 ALIGN=MIDDLE  ALT=\"9x%5E2%2Bhighlight%283x%2B-18x%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=9x%5E2%2B3x-18x-6 ALIGN=MIDDLE  ALT=\"9x%5E2%2B3x-18x-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=%289x%5E2%2B3x%29%2B%28-18x-6%29 ALIGN=MIDDLE  ALT=\"%289x%5E2%2B3x%29%2B%28-18x-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=3x%283x%2B1%29-6%283x%2B1%29 ALIGN=MIDDLE  ALT=\"3x%283x%2B1%29-6%283x%2B1%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=3x ALIGN=MIDDLE  ALT=\"3x\"   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=-6 ALIGN=MIDDLE  ALT=\"-6\"   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=%283x-6%29%283x%2B1%29 ALIGN=MIDDLE  ALT=\"%283x-6%29%283x%2B1%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=3x%2B1 ALIGN=MIDDLE  ALT=\"3x%2B1\"   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=9x%5E2%2B3x-18x-6 ALIGN=MIDDLE  ALT=\"9x%5E2%2B3x-18x-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x-6%29%283x%2B1%29 ALIGN=MIDDLE  ALT=\"%283x-6%29%283x%2B1%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=9x%5E2-15x-6 ALIGN=MIDDLE  ALT=\"9x%5E2-15x-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x-6%29%283x%2B1%29 ALIGN=MIDDLE  ALT=\"%283x-6%29%283x%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9x%5E2-15x-6 ALIGN=MIDDLE  ALT=\"9x%5E2-15x-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9x%5E2%2B3x-18x-6 ALIGN=MIDDLE  ALT=\"9x%5E2%2B3x-18x-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=9x%5E2-15x-6 ALIGN=MIDDLE  ALT=\"9x%5E2-15x-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x-6%29%283x%2B1%29 ALIGN=MIDDLE  ALT=\"%283x-6%29%283x%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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