document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=230532>Question 230532</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me solve this one, thanks in advance.\r<BR>\n" );
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document.write( "x^2/3-x^1/3-20=0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #170727 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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=170727\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The key to the simple solution for this equation is to recognize that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E%282%2F3%29+=+%28x%5E%281%2F3%29%29%5E2 ALIGN=MIDDLE  ALT=\"x%5E%282%2F3%29+=+%28x%5E%281%2F3%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Once we see this, we can recognize that your equation is a quadratic equation in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E%281%2F3%29 ALIGN=MIDDLE  ALT=\"x%5E%281%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and that this quadratic equation will factor fairly easily:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E%281%2F3%29+-+5%29%28x%5E%281%2F3%29+%2B+4%29+=+0 ALIGN=MIDDLE  ALT=\"%28x%5E%281%2F3%29+-+5%29%28x%5E%281%2F3%29+%2B+4%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(If, after thinking about this, you still have trouble seeing this, see below for a way to solve this that is longer but may be clearer.)<BR>\n" );
document.write( "By the Zero Product Property we know that this product is zero only if one of the factors is zero. So:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E%281%2F3%29+-+5+=+0 ALIGN=MIDDLE  ALT=\"x%5E%281%2F3%29+-+5+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E%281%2F3%29+%2B+4+=+0 ALIGN=MIDDLE  ALT=\"x%5E%281%2F3%29+%2B+4+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Solving these we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E%281%2F3%29+=+5 ALIGN=MIDDLE  ALT=\"x%5E%281%2F3%29+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E%281%2F3%29+=+-4 ALIGN=MIDDLE  ALT=\"x%5E%281%2F3%29+=+-4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Cubing both sides:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+125 ALIGN=MIDDLE  ALT=\"x+=+125\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-64 ALIGN=MIDDLE  ALT=\"x+=+-64\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "A longer, perhaps clearer, solution. We still will be using <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E%282%2F3%29+=+x%5E%281%2F3%29%5E2 ALIGN=MIDDLE  ALT=\"x%5E%282%2F3%29+=+x%5E%281%2F3%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. But this time we will use a temporary variable to make things clearer.<BR>\n" );
document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q+=+x%5E%281%2F3%29 ALIGN=MIDDLE  ALT=\"q+=+x%5E%281%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q%5E2+=+%28x%5E%281%2F3%29%29%5E2+=+x%5E%282%2F3%29 ALIGN=MIDDLE  ALT=\"q%5E2+=+%28x%5E%281%2F3%29%29%5E2+=+x%5E%282%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now we'll substitute these into your equation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q%5E2+-+q+-+20+=+0 ALIGN=MIDDLE  ALT=\"q%5E2+-+q+-+20+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This is clearly a quadratic equation and the factoring is more obvious:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28q-5%29%28q%2B4%29=0 ALIGN=MIDDLE  ALT=\"%28q-5%29%28q%2B4%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Solving this we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q-5+=+0 ALIGN=MIDDLE  ALT=\"q-5+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q%2B4=0 ALIGN=MIDDLE  ALT=\"q%2B4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q+=+5 ALIGN=MIDDLE  ALT=\"q+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q+=+-4 ALIGN=MIDDLE  ALT=\"q+=+-4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "At this point we have solved for q. But we want to solve for x. So we substitute back in for q:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E%281%2F3%29+=+5 ALIGN=MIDDLE  ALT=\"x%5E%281%2F3%29+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E%281%2F3%29+=+-4 ALIGN=MIDDLE  ALT=\"x%5E%281%2F3%29+=+-4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "To solve for x we cube both sides of this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+125 ALIGN=MIDDLE  ALT=\"x+=+125\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-64 ALIGN=MIDDLE  ALT=\"x+=+-64\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "If you have to use the temporary variables to solve problems like this, it would be good, after a while, to wean yourself off the temporary variables.<BR>\n" );
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