document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1187475>Question 1187475</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 12(x^2+1/x^2)-56(x+1/x)+89=0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #818445 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13198)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=818445\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Fun problem...!<br><BR>\n" );
document.write( "Here is the key:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B1%2Fx%5E2+=+%28x%5E2%2B2%2B1%2Fx%5E2%29-2+=+%28x%2B1%2Fx%29%5E2-2 ALIGN=MIDDLE  ALT=\"x%5E2%2B1%2Fx%5E2+=+%28x%5E2%2B2%2B1%2Fx%5E2%29-2+=+%28x%2B1%2Fx%29%5E2-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Use that to rewrite the first expression in parentheses and you end up with a quadratic equation with \"x+1/x\" as the \"variable\".<br><BR>\n" );
document.write( "Solve that equation for x+1/x and you end up solving two quadratic equations that each have two real solutions, giving you four solutions to the given equation.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%28x%5E2%2B1%2Fx%5E2%29-56%28x%2B1%2Fx%29%2B89=0 ALIGN=MIDDLE  ALT=\"12%28x%5E2%2B1%2Fx%5E2%29-56%28x%2B1%2Fx%29%2B89=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%28x%5E2%2B2%2B1%2Fx%5E2%29-12%282%29-56%28x%2B1%2Fx%29%2B89=0 ALIGN=MIDDLE  ALT=\"12%28x%5E2%2B2%2B1%2Fx%5E2%29-12%282%29-56%28x%2B1%2Fx%29%2B89=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%28x%2B1%2Fx%29%5E2-24-56%28x%2B1%2Fx%29%2B89=0 ALIGN=MIDDLE  ALT=\"12%28x%2B1%2Fx%29%5E2-24-56%28x%2B1%2Fx%29%2B89=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%28x%2B1%2Fx%29%5E2-56%28x%2B1%2Fx%29%2B65=0 ALIGN=MIDDLE  ALT=\"12%28x%2B1%2Fx%29%5E2-56%28x%2B1%2Fx%29%2B65=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "That's a quadratic equation in the \"variable\" x+1/x -- solve it by factoring.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%28x%2B1%2Fx%29-5%29%286%28x%2B1%2Fx%29-13%29=0 ALIGN=MIDDLE  ALT=\"%282%28x%2B1%2Fx%29-5%29%286%28x%2B1%2Fx%29-13%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B1%2Fx=5%2F2 ALIGN=MIDDLE  ALT=\"x%2B1%2Fx=5%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B1%2Fx=13%2F6 ALIGN=MIDDLE  ALT=\"x%2B1%2Fx=13%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Solve each of those equations....<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B1%2Fx=5%2F2 ALIGN=MIDDLE  ALT=\"x%2B1%2Fx=5%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2B2%2Fx=5 ALIGN=MIDDLE  ALT=\"2x%2B2%2Fx=5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-5%2B2%2Fx=0 ALIGN=MIDDLE  ALT=\"2x-5%2B2%2Fx=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2-5x%2B2=0 ALIGN=MIDDLE  ALT=\"2x%5E2-5x%2B2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x-1%29%28x-2%29=0 ALIGN=MIDDLE  ALT=\"%282x-1%29%28x-2%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1%2F2 ALIGN=MIDDLE  ALT=\"x=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2 ALIGN=MIDDLE  ALT=\"x=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "There are two solutions; note that they are reciprocals of each other.  That makes sense, since the equation can be written as a quadratic with \"variable\" (x+1/x) -- if x=2 then 1/x=1/2; and if x=1/2 then 1/x=2.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B1%2Fx=13%2F6 ALIGN=MIDDLE  ALT=\"x%2B1%2Fx=13%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%2B6%2Fx=13 ALIGN=MIDDLE  ALT=\"6x%2B6%2Fx=13\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x-13%2B6%2Fx=0 ALIGN=MIDDLE  ALT=\"6x-13%2B6%2Fx=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%5E2-13x%2B6=0 ALIGN=MIDDLE  ALT=\"6x%5E2-13x%2B6=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x-3%29%283x-2%29=0 ALIGN=MIDDLE  ALT=\"%282x-3%29%283x-2%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3%2F2 ALIGN=MIDDLE  ALT=\"x=3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2%2F3 ALIGN=MIDDLE  ALT=\"x=2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "There are two more solutions which are also reciprocals of each other.<br><BR>\n" );
document.write( "ANSWER: There are 4 solutions: x=2 or x=1/2; and x=3/2 and x=2/3<br><BR>\n" );
document.write( "Here is a graph of the given equation, showing the four zeros at 1/2, 2/3, 3/2, and 2.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C200%2C-1%2C4%2C-1%2C1%2C12%28x%5E2%2B1%2Fx%5E2%29-56%28x%2B1%2Fx%29%2B89%29 ALIGN=MIDDLE  ALT=\"graph%28600%2C200%2C-1%2C4%2C-1%2C1%2C12%28x%5E2%2B1%2Fx%5E2%29-56%28x%2B1%2Fx%29%2B89%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );