document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1184166>Question 1184166</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve algebraic expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a%5E6+-12a%5E4+%2B+48a%5E2+-+61+=+0+ ALIGN=MIDDLE  ALT=\"+a%5E6+-12a%5E4+%2B+48a%5E2+-+61+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" > and find the real roots. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #814696 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</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=814696\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+f%28a%29+=+a%5E6+-12a%5E4+%2B+48a%5E2+-+61+ ALIGN=MIDDLE  ALT=\"+f%28a%29+=+a%5E6+-12a%5E4+%2B+48a%5E2+-+61+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Graphing the function will help us see if our results are reasonable.  Here is a graph:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-3%2C3%2C-20%2C20%2Cx%5E6-12x%5E4%2B48x%5E2-61%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C-3%2C3%2C-20%2C20%2Cx%5E6-12x%5E4%2B48x%5E2-61%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Note the graph is even (symmetrical with respect to the y-axis, because every exponent in the polynomial is even.<br><BR>\n" );
document.write( "There is no general method for solving a 6th degree polynomial equation.<br><BR>\n" );
document.write( "By far the easiest way to find the APPROXIMATE real roots is by graphing the function on a graphing calculator and finding the zeros from the graph.<br><BR>\n" );
document.write( "My TI-83 shows the real roots to be +/- 1.599296855 to 9 decimal places, which is consistent with the graph.<br><BR>\n" );
document.write( "You can get that same result using Newton's Method.  Look on the internet for a video on how to use Newton's Method on a TI83 or TI84 calculator.<br><BR>\n" );
document.write( "I would consider doing that very worthwhile, since Newton's Method is a very powerful tool.<br><BR>\n" );
document.write( "Finally, there is a way to find the exact real roots for this example.<br><BR>\n" );
document.write( "Since the function is even,  let y=a^2 and write the equation as<br><BR>\n" );
document.write( "x^3-12x^2+48x-61=0<br><BR>\n" );
document.write( "Then look again on the internet and find the formula for the general solution of a cubic equation and use it; then remember that you are solving for a=sqrt(x), not for x.<br><BR>\n" );
document.write( "It is a very ugly formula, which I have never tried to use; and I have no inclination to try it on your example.  So if you want to find the exact real zeros, I say have fun...!<br><BR>\n" );
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