document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=78943>Question 78943</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 12X^2-41X+24 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #56661 by </B><A HREF=/tutors/aboutme.mpl?userid=mathdoc314><B>mathdoc314(58)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=mathdoc314><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=56661\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> That is not a problem.  It is an expression.  There are many things that can be done with an expression.  Next time I hope you will ask the question more completely and clearly!\r<BR>\n" );
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document.write( "Since you are letting me guess what is supposed to be done with it, I could graph it or solve for it = 0, but I think maybe you want me to factor it\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+12+x%5E2+-+41+x+%2B+24+ ALIGN=MIDDLE  ALT=\"+12+x%5E2+-+41+x+%2B+24+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This is going to be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28ax+%2B+b%29+%28cx+%2B+d%29+ ALIGN=MIDDLE  ALT=\"+%28ax+%2B+b%29+%28cx+%2B+d%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "It multiplies out as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+ac+%2A+x%5E2+%2B+%28a%2Ad+%2B+b%2Ac%29+x+%2B+%28b%2Ad%29+ ALIGN=MIDDLE  ALT=\"+ac+%2A+x%5E2+%2B+%28a%2Ad+%2B+b%2Ac%29+x+%2B+%28b%2Ad%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This must be equal to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+12+x%5E2+-+41+x+%2B+24+ ALIGN=MIDDLE  ALT=\"+12+x%5E2+-+41+x+%2B+24+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Therefore it must be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+ac=12+ ALIGN=MIDDLE  ALT=\"+ac=12+\"   DISABLED_style= \"max-width:90%; height:auto;\" >,  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2Ad+%2B+b%2Ac%29+=+-41 ALIGN=MIDDLE  ALT=\"%28a%2Ad+%2B+b%2Ac%29+=+-41\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28b%2Ad%29+=+24+ ALIGN=MIDDLE  ALT=\"+%28b%2Ad%29+=+24+\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "and the goal is to discover right numbers to use for a, b, c, d.<BR>\n" );
document.write( "I will try to find an answer with whole numbers (no fractions or square roots yet).\r<BR>\n" );
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document.write( "a*c = 12 <BR>\n" );
document.write( "some possibilities are<BR>\n" );
document.write( "(a=1,c=12), (a=2,c=6), (a=3,c=4), (a=4,c=3), (a=6,c=2), or (a=12, c=1).  \r<BR>\n" );
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document.write( "b*d = 24 <BR>\n" );
document.write( "some possibilities are<BR>\n" );
document.write( "(b=-1,d=-24), (b=-2,d=-12), (b=-3,d=-8), (b=-4,d=-6)\r<BR>\n" );
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document.write( "The next goal is to find the right combination of those possibilities to get a*d + b*c = -41 \r<BR>\n" );
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document.write( "This can be done slowly by trying out a lot of combinations, but then you probably get tired of it and hopefully notice some patterns.\r<BR>\n" );
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document.write( "What has to happen to get -41\r<BR>\n" );
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document.write( "One thing is that a*d has to be odd or b*c has to be odd.  This eliminates a lot of combinations.  You can't have a and c both be even, and you can't have b and d both be even.  Also you cant have a big number for a and d at the same time because it gets too far over -41 if you have things like 12*8 in it.\r<BR>\n" );
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document.write( "It turns out -41 is the smallest odd term you can get\r<BR>\n" );
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document.write( "Here are some of the details\r<BR>\n" );
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document.write( "You can work with these:(a=1,c=12), (a=3,c=4), (a=4,c=3), (a=12, c=1)<BR>\n" );
document.write( "and (b=-1,d=-24), (b=-3,d=-8)\r<BR>\n" );
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document.write( "(a=4,c=3,b=-1,d=-24) -> ad+bc = -96-3 = -99<BR>\n" );
document.write( "(a=4,c=3,b=-3,d=-8) -> ad+bc = -32 - 9 = -41    <-- this one is the answer<BR>\n" );
document.write( "(a=12,c=1,b=-1,d=-24) -> ad+bd = -288 - 1 = -289<BR>\n" );
document.write( "(a=12,c=1,b=-3,d=-8) -> ad+bd = -96 - 3 = -99\r<BR>\n" );
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document.write( "So it factors as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%284x+-+3%29%283x+-+8%29+=+%2812x%5E2+-+41+x+%2B+24%29+ ALIGN=MIDDLE  ALT=\"+%284x+-+3%29%283x+-+8%29+=+%2812x%5E2+-+41+x+%2B+24%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > and I am glad that is over.  There are faster answers like completing the square and the quadratic formula.  I hope this is the question you wanted to ask.  Next time please say what you want the tutor to do.\r<BR>\n" );
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