document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=116758>Question 116758</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/049510552X/algebrahomeworkh>Intermediate Algebra</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> Please help me solve this equation.  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9x%5E2-17x%5E4%2B21x%5E5 ALIGN=MIDDLE  ALT=\"9x%5E2-17x%5E4%2B21x%5E5\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #87123 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=87123\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is not an equation.  This is a polynomial expression.  As it stands, there is no solution -- you can't 'solve' a polynomial expression. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So, let's presume that you actually meant to say 'Please solve: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9x%5E2-17x%5E4%2B21x%5E5=0 ALIGN=MIDDLE  ALT=\"9x%5E2-17x%5E4%2B21x%5E5=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The first thing you can do is factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > out of the left side leaving you with:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2821x%5E3-17x%5E2%2B9%29=0 ALIGN=MIDDLE  ALT=\"x%5E2%2821x%5E3-17x%5E2%2B9%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "That gives us two of the five roots directly, namely <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "From here on it gets mighty ugly.   Looking at a graph of the cubic factor, we can see that we have one real root, slightly smaller than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%281%2F2%29 ALIGN=MIDDLE  ALT=\"-%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and a conjugate pair of complex roots.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C600%2C-4%2C4%2C-20%2C20%2C21x%5E3-17x%5E2%2B9%29 ALIGN=MIDDLE  ALT=\"graph%28600%2C600%2C-4%2C4%2C-20%2C20%2C21x%5E3-17x%5E2%2B9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "My recommendation, to the extent that your instructor will be satisfied with a reasonable approximation, is to go to a website that has a general cubic equation solver.  When you use such a thing, just be sure to remember that the c coefficient in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E3%2Bbx%5E2%2Bcx%2Bd=0 ALIGN=MIDDLE  ALT=\"ax%5E3%2Bbx%5E2%2Bcx%2Bd=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > has a zero value in this problem.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "There is another possibility that occurred to me -- you may have a typo in your polynomial.  If you actually meant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9x%5E3-17x%5E4%2B21x%5E5 ALIGN=MIDDLE  ALT=\"9x%5E3-17x%5E4%2B21x%5E5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9x%5E3-17x%5E4%2B21x%5E5=0 ALIGN=MIDDLE  ALT=\"9x%5E3-17x%5E4%2B21x%5E5=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > becomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3%2821x%5E2-17x%2B9%29 ALIGN=MIDDLE  ALT=\"x%5E3%2821x%5E2-17x%2B9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and we are left with a much easier to handle quadratic.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " <TABLE  BORDER=1>\n" );
document.write( "\n" );
document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=quadratic.solver>SOLVE quadratic equation with variable</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=21x%5E2%2B-17x%2B9+=+0 ALIGN=MIDDLE  ALT=\"21x%5E2%2B-17x%2B9+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) has the following solutons:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For these solutions to exist, the <U>discriminant</U> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > should not be a negative number.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First, we need to compute the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=%28-17%29%5E2-4%2A21%2A9=-467 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-17%29%5E2-4%2A21%2A9=-467\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  The discriminant -467 is less than zero. That means that there are no solutions among real numbers.<BR><BR>\n" );
document.write( "  If you are a student of advanced school algebra and are aware about <u>imaginary numbers<u>, read on.<BR><BR>\n" );
document.write( "  <BR>\n" );
document.write( "      In the field of imaginary numbers, the square root of -467 is + or - <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28+467%29+=+21.6101827849743 ALIGN=MIDDLE  ALT=\"sqrt%28+467%29+=+21.6101827849743\"   DISABLED_style= \"max-width:90%; height:auto;\" >. <BR>\n" );
document.write( "  <BR>\n" );
document.write( "      The solution is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B12%5D+=+%28--17%2B-+i%2Asqrt%28+-467+%29%29%2F2%5C21+=++%28--17%2B-+i%2A21.6101827849743%29%2F2%5C21+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Here's your graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+21%2Ax%5E2%2B-17%2Ax%2B9+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+21%2Ax%5E2%2B-17%2Ax%2B9+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Not nearly as messy, even though the roots are complex.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Hope this helps.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );