document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=417524>Question 417524</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do you factor: 4x squared - 17x + 4 ? \r<BR>\n" );
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document.write( "This factoring thing doesn't make sense to me at all.... :( </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #292296 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=292296\">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=4x%5E2-17x%2B4 ALIGN=MIDDLE  ALT=\"4x%5E2-17x%2B4\"   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=4x%5E2 ALIGN=MIDDLE  ALT=\"4x%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=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > where the coefficients are 4 and 4 respectively.\r<BR>\n" );
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document.write( "Now multiply the first coefficient 4 and the last coefficient 4 to get 16. Now what two numbers multiply to 16 and add to the  middle coefficient -17? Let's list all of the factors of 16:\r<BR>\n" );
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document.write( "Factors of 16:\r<BR>\n" );
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document.write( "1,2,4,8\r<BR>\n" );
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document.write( "-1,-2,-4,-8 ...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 16\r<BR>\n" );
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document.write( "1*16\r<BR>\n" );
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document.write( "2*8\r<BR>\n" );
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document.write( "4*4\r<BR>\n" );
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document.write( "(-1)*(-16)\r<BR>\n" );
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document.write( "(-2)*(-8)\r<BR>\n" );
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document.write( "(-4)*(-4)\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 -17? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -17\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\">16</TD><TD >1+16=17</TD></tr><tr><TD  align=\"center\">2</TD><TD  align=\"center\">8</TD><TD >2+8=10</TD></tr><tr><TD  align=\"center\">4</TD><TD  align=\"center\">4</TD><TD >4+4=8</TD></tr><tr><TD  align=\"center\">-1</TD><TD  align=\"center\">-16</TD><TD >-1+(-16)=-17</TD></tr><tr><TD  align=\"center\">-2</TD><TD  align=\"center\">-8</TD><TD >-2+(-8)=-10</TD></tr><tr><TD  align=\"center\">-4</TD><TD  align=\"center\">-4</TD><TD >-4+(-4)=-8</TD></tr>\n" );
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document.write( "From this list we can see that -1 and -16 add up to -17 and multiply to 16\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=4x%5E2-17x%2B4 ALIGN=MIDDLE  ALT=\"4x%5E2-17x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-17x ALIGN=MIDDLE  ALT=\"-17x\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1x%2B-16x ALIGN=MIDDLE  ALT=\"-1x%2B-16x\"   DISABLED_style= \"max-width:90%; height:auto;\" > (notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1x%2B-16x ALIGN=MIDDLE  ALT=\"-1x%2B-16x\"   DISABLED_style= \"max-width:90%; height:auto;\" > adds up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-17x ALIGN=MIDDLE  ALT=\"-17x\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So it is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-17x ALIGN=MIDDLE  ALT=\"-17x\"   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=4x%5E2%2Bhighlight%28-1x%2B-16x%29%2B4 ALIGN=MIDDLE  ALT=\"4x%5E2%2Bhighlight%28-1x%2B-16x%29%2B4\"   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=4x%5E2-1x-16x%2B4 ALIGN=MIDDLE  ALT=\"4x%5E2-1x-16x%2B4\"   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=%284x%5E2-1x%29%2B%28-16x%2B4%29 ALIGN=MIDDLE  ALT=\"%284x%5E2-1x%29%2B%28-16x%2B4%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%284x-1%29-4%284x-1%29 ALIGN=MIDDLE  ALT=\"x%284x-1%29-4%284x-1%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=-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=%28x-4%29%284x-1%29 ALIGN=MIDDLE  ALT=\"%28x-4%29%284x-1%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=4x-1 ALIGN=MIDDLE  ALT=\"4x-1\"   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=4x%5E2-1x-16x%2B4 ALIGN=MIDDLE  ALT=\"4x%5E2-1x-16x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-4%29%284x-1%29 ALIGN=MIDDLE  ALT=\"%28x-4%29%284x-1%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=4x%5E2-17x%2B4 ALIGN=MIDDLE  ALT=\"4x%5E2-17x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-4%29%284x-1%29 ALIGN=MIDDLE  ALT=\"%28x-4%29%284x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2-17x%2B4 ALIGN=MIDDLE  ALT=\"4x%5E2-17x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2-1x-16x%2B4 ALIGN=MIDDLE  ALT=\"4x%5E2-1x-16x%2B4\"   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=4x%5E2-17x%2B4 ALIGN=MIDDLE  ALT=\"4x%5E2-17x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-4%29%284x-1%29 ALIGN=MIDDLE  ALT=\"%28x-4%29%284x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Jim </SPAN>\n" );
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