document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=812942>Question 812942</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve<BR>\n" );
document.write( "a) x^(1/2) - x^(1/4) - 2 = 0\r<BR>\n" );
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document.write( "Can you please help me out? Thanks so much in advance:) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #489447 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=489447\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E%281%2F2%29-x%5E%281%2F4%29-2=0 ALIGN=MIDDLE  ALT=\"x%5E%281%2F2%29-x%5E%281%2F4%29-2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since the first exponent, 1/2, is twice the second exponent, 1/4, then this equation is what is known as \"quadratic form\". These types of equations can be solved using techniques that are used to solve regular quadratic equations.<br><BR>\n" );
document.write( "Until you have had some practice with these equations they can be difficult. It can help to use a temporary variable. Set it to the variable and exponent in the middle:<BR>\n" );
document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q+=+x%5E%281%2F4%29 ALIGN=MIDDLE  ALT=\"q+=+x%5E%281%2F4%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%2F4%29%29%5E2+=+x%5E%281%2F2%29 ALIGN=MIDDLE  ALT=\"q%5E2+=+%28x%5E%281%2F4%29%29%5E2+=+x%5E%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Substituting these into your equation we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q%5E2+-q+-2+=+0 ALIGN=MIDDLE  ALT=\"q%5E2+-q+-2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This now actually looks like a quadratic equation. It will factor:<BR>\n" );
document.write( "(q+1)(q-2) = 0<BR>\n" );
document.write( "From the Zero Product Property:<BR>\n" );
document.write( "q+1 = 0 or q-2 = 0<BR>\n" );
document.write( "Solving these we get:<BR>\n" );
document.write( "q = -1 or q = 2<br><BR>\n" );
document.write( "We have solved for our made-up variable q. But we really want solutions for x. At this point we substitute back in for the q:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E%281%2F4%29+=+-1 ALIGN=MIDDLE  ALT=\"x%5E%281%2F4%29+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > or<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E%281%2F4%29+=+2 ALIGN=MIDDLE  ALT=\"x%5E%281%2F4%29+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "And last of all we solve for x.<br><BR>\n" );
document.write( "The first equation says that a 4th root is equal to a negative number. 4th roots (without a minus in front) must be positive so we will not get any (real) solutions from the first equation.<br><BR>\n" );
document.write( "To solve the second equation we just raise each side to the 4th power:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E%281%2F4%29%29%5E4+=+2%5E4 ALIGN=MIDDLE  ALT=\"%28x%5E%281%2F4%29%29%5E4+=+2%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which simplifies to:<BR>\n" );
document.write( "x = 16<br><BR>\n" );
document.write( "Since we raised each side of an equation by an even power we should check. Use the original equation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E%281%2F2%29-x%5E%281%2F4%29-2=0 ALIGN=MIDDLE  ALT=\"x%5E%281%2F2%29-x%5E%281%2F4%29-2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Checking x = 16:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2816%29%5E%281%2F2%29-%2816%29%5E%281%2F4%29-2=0 ALIGN=MIDDLE  ALT=\"%2816%29%5E%281%2F2%29-%2816%29%5E%281%2F4%29-2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since 1/2 as an exponent means square root and an exponent of 1/4 means 4th root and since we know both the square and 4th roots of 16 this simplifies to:<BR>\n" );
document.write( "4 - 2 - 2 = 0<BR>\n" );
document.write( "which further simplifies to:<BR>\n" );
document.write( "0 = 0 Check!<br><BR>\n" );
document.write( "P.S. Once you get some practice with these you will no longer need the temporary variable. You will start seeing how to go directly from<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E%281%2F2%29-x%5E%281%2F4%29-2=0 ALIGN=MIDDLE  ALT=\"x%5E%281%2F2%29-x%5E%281%2F4%29-2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E%281%2F4%29%2B1%29%28x%5E%281%2F4%29-2%29+=+0 ALIGN=MIDDLE  ALT=\"%28x%5E%281%2F4%29%2B1%29%28x%5E%281%2F4%29-2%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E%281%2F4%29%2B1+=+0 ALIGN=MIDDLE  ALT=\"x%5E%281%2F4%29%2B1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E%281%2F4%29-2+=+0 ALIGN=MIDDLE  ALT=\"x%5E%281%2F4%29-2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "etc. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );