document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=117621>Question 117621</A>: <I><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "9x^2+18x+8 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #85614 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=85614\">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%2B18x%2B8 ALIGN=MIDDLE  ALT=\"9x%5E2%2B18x%2B8\"   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=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" > where the coefficients are 9 and 8 respectively.\r<BR>\n" );
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document.write( "Now multiply the first coefficient 9 and the last coefficient 8 to get 72. Now what two numbers multiply to 72 and add to the  middle coefficient 18? Let's list all of the factors of 72:\r<BR>\n" );
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document.write( "Factors of 72:\r<BR>\n" );
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document.write( "1,2,3,4,6,8,9,12,18,24,36,72\r<BR>\n" );
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document.write( "-1,-2,-3,-4,-6,-8,-9,-12,-18,-24,-36,-72 ...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 72\r<BR>\n" );
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document.write( "1*72\r<BR>\n" );
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document.write( "2*36\r<BR>\n" );
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document.write( "3*24\r<BR>\n" );
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document.write( "4*18\r<BR>\n" );
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document.write( "6*12\r<BR>\n" );
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document.write( "8*9\r<BR>\n" );
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document.write( "(-1)*(-72)\r<BR>\n" );
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document.write( "(-2)*(-36)\r<BR>\n" );
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document.write( "(-3)*(-24)\r<BR>\n" );
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document.write( "(-4)*(-18)\r<BR>\n" );
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document.write( "(-6)*(-12)\r<BR>\n" );
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document.write( "(-8)*(-9)\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 18? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 18\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\">72</TD><TD >1+72=73</TD></tr><tr><TD  align=\"center\">2</TD><TD  align=\"center\">36</TD><TD >2+36=38</TD></tr><tr><TD  align=\"center\">3</TD><TD  align=\"center\">24</TD><TD >3+24=27</TD></tr><tr><TD  align=\"center\">4</TD><TD  align=\"center\">18</TD><TD >4+18=22</TD></tr><tr><TD  align=\"center\">6</TD><TD  align=\"center\">12</TD><TD >6+12=18</TD></tr><tr><TD  align=\"center\">8</TD><TD  align=\"center\">9</TD><TD >8+9=17</TD></tr><tr><TD  align=\"center\">-1</TD><TD  align=\"center\">-72</TD><TD >-1+(-72)=-73</TD></tr><tr><TD  align=\"center\">-2</TD><TD  align=\"center\">-36</TD><TD >-2+(-36)=-38</TD></tr><tr><TD  align=\"center\">-3</TD><TD  align=\"center\">-24</TD><TD >-3+(-24)=-27</TD></tr><tr><TD  align=\"center\">-4</TD><TD  align=\"center\">-18</TD><TD >-4+(-18)=-22</TD></tr><tr><TD  align=\"center\">-6</TD><TD  align=\"center\">-12</TD><TD >-6+(-12)=-18</TD></tr><tr><TD  align=\"center\">-8</TD><TD  align=\"center\">-9</TD><TD >-8+(-9)=-17</TD></tr>\n" );
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document.write( "From this list we can see that 6 and 12 add up to 18 and multiply to 72\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%2B18x%2B8 ALIGN=MIDDLE  ALT=\"9x%5E2%2B18x%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" >, replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18x ALIGN=MIDDLE  ALT=\"18x\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%2B12x ALIGN=MIDDLE  ALT=\"6x%2B12x\"   DISABLED_style= \"max-width:90%; height:auto;\" > (notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%2B12x ALIGN=MIDDLE  ALT=\"6x%2B12x\"   DISABLED_style= \"max-width:90%; height:auto;\" > adds up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18x ALIGN=MIDDLE  ALT=\"18x\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So it is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18x ALIGN=MIDDLE  ALT=\"18x\"   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%286x%2B12x%29%2B8 ALIGN=MIDDLE  ALT=\"9x%5E2%2Bhighlight%286x%2B12x%29%2B8\"   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%2B6x%2B12x%2B8 ALIGN=MIDDLE  ALT=\"9x%5E2%2B6x%2B12x%2B8\"   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%2B6x%29%2B%2812x%2B8%29 ALIGN=MIDDLE  ALT=\"%289x%5E2%2B6x%29%2B%2812x%2B8%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%2B2%29%2B4%283x%2B2%29 ALIGN=MIDDLE  ALT=\"3x%283x%2B2%29%2B4%283x%2B2%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=4 ALIGN=MIDDLE  ALT=\"4\"   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%2B4%29%283x%2B2%29 ALIGN=MIDDLE  ALT=\"%283x%2B4%29%283x%2B2%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%2B2 ALIGN=MIDDLE  ALT=\"3x%2B2\"   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%2B6x%2B12x%2B8 ALIGN=MIDDLE  ALT=\"9x%5E2%2B6x%2B12x%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x%2B4%29%283x%2B2%29 ALIGN=MIDDLE  ALT=\"%283x%2B4%29%283x%2B2%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%2B18x%2B8 ALIGN=MIDDLE  ALT=\"9x%5E2%2B18x%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x%2B4%29%283x%2B2%29 ALIGN=MIDDLE  ALT=\"%283x%2B4%29%283x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9x%5E2%2B18x%2B8 ALIGN=MIDDLE  ALT=\"9x%5E2%2B18x%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9x%5E2%2B6x%2B12x%2B8 ALIGN=MIDDLE  ALT=\"9x%5E2%2B6x%2B12x%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "-------------------------------\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%2B18x%2B8 ALIGN=MIDDLE  ALT=\"9x%5E2%2B18x%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x%2B4%29%283x%2B2%29 ALIGN=MIDDLE  ALT=\"%283x%2B4%29%283x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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