document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=108982>Question 108982</A>: <I><SPAN STYLE=\"word-break: break-all;\"> There was a party and everyone shook everyone's hands. There were 66 handshakes!! How many people were at the party???????????<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #79462 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20849)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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/cgi-bin/show-face.mpl?user=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=79462\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> There was a party and everyone shook everyone's hands. There were 66 handshakes!! How many people were at the party?\r<BR>\n" );
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document.write( "we know that:\r<BR>\n" );
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document.write( "with two people (x and y), there is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=one ALIGN=MIDDLE  ALT=\"one\"   DISABLED_style= \"max-width:90%; height:auto;\" > handshake; (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "with three people (x,y, and z), there are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=three ALIGN=MIDDLE  ALT=\"three\"   DISABLED_style= \"max-width:90%; height:auto;\" > handshakes; x with y and z, and y with z\r<BR>\n" );
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document.write( "with four people (x, y, z, and e), there are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=six ALIGN=MIDDLE  ALT=\"six\"   DISABLED_style= \"max-width:90%; height:auto;\" > handshakes; x with y, z, and e, then  y with z and e, then  z with e\r<BR>\n" );
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document.write( "In general:\r<BR>\n" );
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document.write( " with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%2B1 ALIGN=MIDDLE  ALT=\"n%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > people, the number of handshakes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=is+-the-+sum ALIGN=MIDDLE  ALT=\"is+-the-+sum\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the\r<BR>\n" );
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document.write( " first <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > consecutive numbers: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B2%2B3%2B+...+%2B+n ALIGN=MIDDLE  ALT=\"1%2B2%2B3%2B+...+%2B+n\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Since this <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum ALIGN=MIDDLE  ALT=\"sum\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%28n%2B1%29%2F2 ALIGN=MIDDLE  ALT=\"n%28n%2B1%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, \r<BR>\n" );
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document.write( "we need to solve the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%0D%0A%0D%0A%28n%2B1%29%2F2+=+66 ALIGN=MIDDLE  ALT=\"n%0D%0A%0D%0A%28n%2B1%29%2F2+=+66\"   DISABLED_style= \"max-width:90%; height:auto;\" > ...multiply both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.=> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%28n+%2B+1%29=+132 ALIGN=MIDDLE  ALT=\"n%28n+%2B+1%29=+132\"   DISABLED_style= \"max-width:90%; height:auto;\" >....=>...<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%5E2+%2B+n+=+132 ALIGN=MIDDLE  ALT=\"n%5E2+%2B+n+=+132\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "this is the quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%5E2%2B+n+-132+=+0 ALIGN=MIDDLE  ALT=\"n%5E2%2B+n+-132+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "use quadratic formula and solve for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   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=x%5B1%2C2%5D=%28-b+%2B-+sqrt+%28b%5E2+-4%2Aa%2Ac+%29%29+%2F+%282%2Aa%29 ALIGN=MIDDLE  ALT=\"x%5B1%2C2%5D=%28-b+%2B-+sqrt+%28b%5E2+-4%2Aa%2Ac+%29%29+%2F+%282%2Aa%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+1 ALIGN=MIDDLE  ALT=\"a+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >}, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+1 ALIGN=MIDDLE  ALT=\"b+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+=+-132 ALIGN=MIDDLE  ALT=\"c+=+-132\"   DISABLED_style= \"max-width:90%; height:auto;\" >, you will have:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%2C2%5D=%28-1+%2B-+sqrt+%281%5E2+-4%2A1%2A%28-132%29+%29%29+%2F+%282%2A1%29 ALIGN=MIDDLE  ALT=\"x%5B1%2C2%5D=%28-1+%2B-+sqrt+%281%5E2+-4%2A1%2A%28-132%29+%29%29+%2F+%282%2A1%29\"   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=x%5B1%2C2%5D=%28-1+%2B-+sqrt+%281+%2B+528+%29%29+%2F+2 ALIGN=MIDDLE  ALT=\"x%5B1%2C2%5D=%28-1+%2B-+sqrt+%281+%2B+528+%29%29+%2F+2\"   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=x%5B1%2C2%5D=%28-1+%2B-+sqrt+%28529+%29%29+%2F+2 ALIGN=MIDDLE  ALT=\"x%5B1%2C2%5D=%28-1+%2B-+sqrt+%28529+%29%29+%2F+2\"   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=x%5B1%2C2%5D=%28-1+%2B-+23+%29+%2F+2 ALIGN=MIDDLE  ALT=\"x%5B1%2C2%5D=%28-1+%2B-+23+%29+%2F+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "we need only positive root:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D=%28-1+%2B+23+%29+%2F+2 ALIGN=MIDDLE  ALT=\"x%5B1%5D=%28-1+%2B+23+%29+%2F+2\"   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=x%5B1%2C2%5D=+22+%2F+2 ALIGN=MIDDLE  ALT=\"x%5B1%2C2%5D=+22+%2F+2\"   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=x%5B1%2C2%5D=+11 ALIGN=MIDDLE  ALT=\"x%5B1%2C2%5D=+11\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " we obtain <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11 ALIGN=MIDDLE  ALT=\"11\"   DISABLED_style= \"max-width:90%; height:auto;\" > as the answer and deduce that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=there-+were+-12-+people-+at-+the-+party ALIGN=MIDDLE  ALT=\"there-+were+-12-+people-+at-+the-+party\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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