document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199136>Question 1199136</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If a polygonal n sides has (n/2)(n - 3) diagonals, how many sides will a polygon with 65 diagonals have? Is there a polygon with 80 diagonals?\r<BR>\n" );
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document.write( "Let me see.\r<BR>\n" );
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document.write( "I gotta set (n/2)(n - 3) equal to 65 and solve for n. The same with 80.\r<BR>\n" );
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document.write( "Yes? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #832848 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=832848\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "n = number of sides, such that n > 3.<BR>\n" );
document.write( "d = number of diagonals of a polygon with n sides<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+%28n%2F2%29%2A%28n-3%29 ALIGN=MIDDLE  ALT=\"d+=+%28n%2F2%29%2A%28n-3%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=d+=+%28n%2A%28n-3%29%29%2F2 ALIGN=MIDDLE  ALT=\"d+=+%28n%2A%28n-3%29%29%2F2\"   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=65+=+%28n%2A%28n-3%29%29%2F2 ALIGN=MIDDLE  ALT=\"65+=+%28n%2A%28n-3%29%29%2F2\"   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=65%2A2+=+n%2A%28n-3%29 ALIGN=MIDDLE  ALT=\"65%2A2+=+n%2A%28n-3%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=130+=+n%5E2-3n ALIGN=MIDDLE  ALT=\"130+=+n%5E2-3n\"   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=n%5E2-3n-130+=+0 ALIGN=MIDDLE  ALT=\"n%5E2-3n-130+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Apply the quadratic formula to solve for n.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"n+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%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=n+=+%28-%28-3%29%2B-sqrt%28%28-3%29%5E2-4%281%29%28-130%29%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"n+=+%28-%28-3%29%2B-sqrt%28%28-3%29%5E2-4%281%29%28-130%29%29%29%2F%282%281%29%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=n+=+%283%2B-sqrt%28529%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"n+=+%283%2B-sqrt%28529%29%29%2F%282%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=n+=+%283%2B-++23%29%2F%282%29 ALIGN=MIDDLE  ALT=\"n+=+%283%2B-++23%29%2F%282%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=n+=+%283%2B23%29%2F%282%29 ALIGN=MIDDLE  ALT=\"n+=+%283%2B23%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+%283-23%29%2F%282%29 ALIGN=MIDDLE  ALT=\"n+=+%283-23%29%2F%282%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=n+=+%2826%29%2F%282%29 ALIGN=MIDDLE  ALT=\"n+=+%2826%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+%28-20%29%2F%282%29 ALIGN=MIDDLE  ALT=\"n+=+%28-20%29%2F%282%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=n+=+13 ALIGN=MIDDLE  ALT=\"n+=+13\"   DISABLED_style= \"max-width:90%; height:auto;\" > or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+-10 ALIGN=MIDDLE  ALT=\"n+=+-10\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Or you can factor like so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%5E2-3n-130+=+0 ALIGN=MIDDLE  ALT=\"n%5E2-3n-130+=+0\"   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=%28n-13%29%28n%2B10%29+=+0 ALIGN=MIDDLE  ALT=\"%28n-13%29%28n%2B10%29+=+0\"   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=n-13+=+0 ALIGN=MIDDLE  ALT=\"n-13+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%2B10+=+0 ALIGN=MIDDLE  ALT=\"n%2B10+=+0\"   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=n+=+13 ALIGN=MIDDLE  ALT=\"n+=+13\"   DISABLED_style= \"max-width:90%; height:auto;\" > or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+-10 ALIGN=MIDDLE  ALT=\"n+=+-10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Ignore the negative value of n. <BR>\n" );
document.write( "It doesn't make sense to have a negative number of sides.\r<BR>\n" );
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document.write( "Therefore, a polygon with n = 13 sides will have d = 65 diagonals.\r<BR>\n" );
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document.write( "If you were to plug in d = 80, then,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+%28n%2A%28n-3%29%29%2F2 ALIGN=MIDDLE  ALT=\"d+=+%28n%2A%28n-3%29%29%2F2\"   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=80+=+%28n%2A%28n-3%29%29%2F2 ALIGN=MIDDLE  ALT=\"80+=+%28n%2A%28n-3%29%29%2F2\"   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=80%2A2+=+n%2A%28n-3%29 ALIGN=MIDDLE  ALT=\"80%2A2+=+n%2A%28n-3%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=n%5E2-3n+=+160 ALIGN=MIDDLE  ALT=\"n%5E2-3n+=+160\"   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=n%5E2-3n-160+=+0 ALIGN=MIDDLE  ALT=\"n%5E2-3n-160+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now use the quadratic formula<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"n+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%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=n+=+%28-%28-3%29%2B-sqrt%28%28-3%29%5E2-4%281%29%28-160%29%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"n+=+%28-%28-3%29%2B-sqrt%28%28-3%29%5E2-4%281%29%28-160%29%29%29%2F%282%281%29%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=n+=+%283%2B-sqrt%28649%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"n+=+%283%2B-sqrt%28649%29%29%2F%282%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=n+=+%283%2B-++25.475478%29%2F%282%29 ALIGN=MIDDLE  ALT=\"n+=+%283%2B-++25.475478%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Each decimal value from here on out is approximate.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+%283%2B25.475478%29%2F%282%29 ALIGN=MIDDLE  ALT=\"n+=+%283%2B25.475478%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+%283-25.475478%29%2F%282%29 ALIGN=MIDDLE  ALT=\"n+=+%283-25.475478%29%2F%282%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=n+=+%2828.475478%29%2F%282%29 ALIGN=MIDDLE  ALT=\"n+=+%2828.475478%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+%28-22.475478%29%2F%282%29 ALIGN=MIDDLE  ALT=\"n+=+%28-22.475478%29%2F%282%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=n+=+14.237739 ALIGN=MIDDLE  ALT=\"n+=+14.237739\"   DISABLED_style= \"max-width:90%; height:auto;\" > or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+-11.237739 ALIGN=MIDDLE  ALT=\"n+=+-11.237739\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We unfortunately do not get a positive integer solution in either case.<BR>\n" );
document.write( "This means it's impossible to have a polygon with exactly 80 diagonals.\r<BR>\n" );
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document.write( "We can see this by constructing a table of values <BR>\n" );
document.write( "n = number of sides<BR>\n" );
document.write( "d = number of diagonals<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >n</TD><TD >4</TD><TD >5</TD><TD >6</TD><TD >7</TD><TD >8</TD><TD >9</TD><TD >10</TD><TD >11</TD><TD >12</TD><TD >13</TD><TD >14</TD><TD >15</TD></tr><tr><TD >d</TD><TD >2</TD><TD >5</TD><TD >9</TD><TD >14</TD><TD >20</TD><TD >27</TD><TD >35</TD><TD >44</TD><TD >54</TD><TD >65</TD><TD >77</TD><TD >90</TD></tr>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "Example scratch work calculation for n = 10:<BR>\n" );
document.write( "d = (n*(n-3))/2 = (10*(10-3))/2 = 35\r<BR>\n" );
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document.write( "The value d = 80 does not exist in the table. The closest two values are d = 77 and d = 90 (for n = 14 and n = 15 respectively).<BR>\n" );
document.write( "Note the solution <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+14.237739 ALIGN=MIDDLE  ALT=\"n+=+14.237739\"   DISABLED_style= \"max-width:90%; height:auto;\" > found earlier is between n = 14 and n = 15.\r<BR>\n" );
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document.write( "Another way to see that we won't have a positive integer solution is to compute the discriminant<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D+=+b%5E2-4ac ALIGN=MIDDLE  ALT=\"D+=+b%5E2-4ac\"   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=D+=+9+%2B+640 ALIGN=MIDDLE  ALT=\"D+=+9+%2B+640\"   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=D+=+649 ALIGN=MIDDLE  ALT=\"D+=+649\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This result is not a perfect square, which means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28D%29+=+sqrt%28649%29 ALIGN=MIDDLE  ALT=\"sqrt%28D%29+=+sqrt%28649%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > isn't an integer.<BR>\n" );
document.write( "Consequently it causes both roots to be non-integer values.<BR>\n" );
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