document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=283083>Question 283083</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How many real numbers are there such that the 5th power of the number is the sum of the 4th and  3rd powers of the number?\r<BR>\n" );
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document.write( "a) 1 b) 2 c) 3 d) 5 e) none of these<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 #205503 by </B><A HREF=/tutors/aboutme.mpl?userid=nabla><B>nabla(475)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=nabla><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=205503\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> So, we want:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E5=x%5E4%2Bx%5E3 ALIGN=MIDDLE  ALT=\"x%5E5=x%5E4%2Bx%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "In other words we are asking when<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E5-x%5E4-x%5E3=0 ALIGN=MIDDLE  ALT=\"x%5E5-x%5E4-x%5E3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >?<BR>\n" );
document.write( "In even more words we are asking when<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3%28x%5E2-x-1%29=0 ALIGN=MIDDLE  ALT=\"x%5E3%28x%5E2-x-1%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >?<BR>\n" );
document.write( "We note that x=0 certainly makes the statement true. Now, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-x-1 ALIGN=MIDDLE  ALT=\"x%5E2-x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > will give two real (irrational) roots. So, strictly speaking there are three real numbers that meet the criterion: {0, 1/2 (1 - Sqrt[5]), 1/2 (1 + Sqrt[5]).\r<BR>\n" );
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document.write( "However, one should note that x=0 occurs as a root three times. </SPAN>\n" );
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