document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=151647>Question 151647</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0618250034/algebrahomeworkh>PreAlgebra</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> I need to find z when z^6=64.  I know the answer is 2 but what are the steps to solve for z? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #111476 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=111476\">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=z%5E6=64 ALIGN=MIDDLE  ALT=\"z%5E6=64\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E6=2%5E6 ALIGN=MIDDLE  ALT=\"z%5E6=2%5E6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite 64 as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E6 ALIGN=MIDDLE  ALT=\"2%5E6\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "From here you can see that the answer is 2, but let's keep going\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%286%2Cz%5E6%29=root%286%2C2%5E6%29 ALIGN=MIDDLE  ALT=\"root%286%2Cz%5E6%29=root%286%2C2%5E6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the 6th root of both sides. This will eliminate the exponent \"6\"\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=2 ALIGN=MIDDLE  ALT=\"z=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the 6th root of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E6 ALIGN=MIDDLE  ALT=\"z%5E6\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get \"z\". Take the 6th root of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E6 ALIGN=MIDDLE  ALT=\"2%5E6\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get 2\r<BR>\n" );
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document.write( "So the solution is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=2 ALIGN=MIDDLE  ALT=\"z=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Here's another way to solve for \"z\"\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E6=64 ALIGN=MIDDLE  ALT=\"z%5E6=64\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E6-64=0 ALIGN=MIDDLE  ALT=\"z%5E6-64=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 64 from both sides.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E6-2%5E6=0 ALIGN=MIDDLE  ALT=\"z%5E6-2%5E6=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite 64 as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E6 ALIGN=MIDDLE  ALT=\"2%5E6\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28z%5E3%29%5E2-%282%5E3%29%5E2=0 ALIGN=MIDDLE  ALT=\"%28z%5E3%29%5E2-%282%5E3%29%5E2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Rewrite <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E6 ALIGN=MIDDLE  ALT=\"z%5E6\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28z%5E3%29%5E2 ALIGN=MIDDLE  ALT=\"%28z%5E3%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Rewrite <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E6 ALIGN=MIDDLE  ALT=\"2%5E6\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%5E3%29%5E2 ALIGN=MIDDLE  ALT=\"%282%5E3%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=z%5E3 ALIGN=MIDDLE  ALT=\"A=z%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B=2%5E3 ALIGN=MIDDLE  ALT=\"B=2%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So we now have\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28A%2BB%29%28A-B%29=0 ALIGN=MIDDLE  ALT=\"%28A%2BB%29%28A-B%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor using the difference of squares formula\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28z%5E3%2B2%5E3%29%28z%5E3-2%5E3%29=0 ALIGN=MIDDLE  ALT=\"%28z%5E3%2B2%5E3%29%28z%5E3-2%5E3%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=z%5E3 ALIGN=MIDDLE  ALT=\"A=z%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B=2%5E3 ALIGN=MIDDLE  ALT=\"B=2%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28z%2B2%29%2A%28z%5E2-2%2Az%2B4%29%29%28z%5E3-2%5E3%29=0 ALIGN=MIDDLE  ALT=\"%28%28z%2B2%29%2A%28z%5E2-2%2Az%2B4%29%29%28z%5E3-2%5E3%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor the first binomial <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E3%2B2%5E3 ALIGN=MIDDLE  ALT=\"z%5E3%2B2%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > using the sum of cubes formula\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28z%2B2%29%2A%28z%5E2-2%2Az%2B4%29%29%28%28z-2%29%2A%28z%5E2%2B2%2Az%2B4%29%29=0 ALIGN=MIDDLE  ALT=\"%28%28z%2B2%29%2A%28z%5E2-2%2Az%2B4%29%29%28%28z-2%29%2A%28z%5E2%2B2%2Az%2B4%29%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor the second binomial <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E3-2%5E3 ALIGN=MIDDLE  ALT=\"z%5E3-2%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > using the sum of cubes formula\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28z%2B2%29%28z-2%29%28z%5E2-2%2Az%2B4%29%28z%5E2%2B2%2Az%2B4%29=0 ALIGN=MIDDLE  ALT=\"%28z%2B2%29%28z-2%29%28z%5E2-2%2Az%2B4%29%28z%5E2%2B2%2Az%2B4%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange the terms\r<BR>\n" );
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document.write( "Now set each factor equal to zero:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%2B2=0 ALIGN=MIDDLE  ALT=\"z%2B2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z-2=0 ALIGN=MIDDLE  ALT=\"z-2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E2-2%2Az%2B4=0 ALIGN=MIDDLE  ALT=\"z%5E2-2%2Az%2B4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E2%2B2%2Az%2B4=0 ALIGN=MIDDLE  ALT=\"z%5E2%2B2%2Az%2B4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So our first two solutions are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=-2 ALIGN=MIDDLE  ALT=\"z=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=2 ALIGN=MIDDLE  ALT=\"z=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now let's solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E2-2%2Az%2B4=0 ALIGN=MIDDLE  ALT=\"z%5E2-2%2Az%2B4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E2-2z%2B4=0 ALIGN=MIDDLE  ALT=\"z%5E2-2z%2B4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation.\r<BR>\n" );
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document.write( "Notice we have a quadratic equation in the form of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=az%5E2%2Bbz%2Bc ALIGN=MIDDLE  ALT=\"az%5E2%2Bbz%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-2 ALIGN=MIDDLE  ALT=\"b=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=4 ALIGN=MIDDLE  ALT=\"c=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's use the quadratic formula to solve for z\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"z+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the quadratic formula\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-%28-2%29+%2B-+sqrt%28+%28-2%29%5E2-4%281%29%284%29+%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"z+=+%28-%28-2%29+%2B-+sqrt%28+%28-2%29%5E2-4%281%29%284%29+%29%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-2 ALIGN=MIDDLE  ALT=\"b=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=4 ALIGN=MIDDLE  ALT=\"c=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%282+%2B-+sqrt%28+%28-2%29%5E2-4%281%29%284%29+%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"z+=+%282+%2B-+sqrt%28+%28-2%29%5E2-4%281%29%284%29+%29%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Negate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2 ALIGN=MIDDLE  ALT=\"-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%282+%2B-+sqrt%28+4-4%281%29%284%29+%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"z+=+%282+%2B-+sqrt%28+4-4%281%29%284%29+%29%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2 ALIGN=MIDDLE  ALT=\"-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%282+%2B-+sqrt%28+4-16+%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"z+=+%282+%2B-+sqrt%28+4-16+%29%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%281%29%284%29 ALIGN=MIDDLE  ALT=\"4%281%29%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16 ALIGN=MIDDLE  ALT=\"16\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%282+%2B-+sqrt%28+-12+%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"z+=+%282+%2B-+sqrt%28+-12+%29%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16 ALIGN=MIDDLE  ALT=\"16\"   DISABLED_style= \"max-width:90%; height:auto;\" > from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-12 ALIGN=MIDDLE  ALT=\"-12\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%282+%2B-+sqrt%28+-12+%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"z+=+%282+%2B-+sqrt%28+-12+%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%282+%2B-+2i%2Asqrt%283%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"z+=+%282+%2B-+2i%2Asqrt%283%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify the square root  (note: If you need help with simplifying square roots, check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)  \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%282%29%2F%282%29+%2B-+%282i%2Asqrt%283%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"z+=+%282%29%2F%282%29+%2B-+%282i%2Asqrt%283%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the fraction.  \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+1+%2B-+sqrt%283%29%2Ai ALIGN=MIDDLE  ALT=\"z+=+1+%2B-+sqrt%283%29%2Ai\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce.  \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+1%2Bsqrt%283%29%2Ai ALIGN=MIDDLE  ALT=\"z+=+1%2Bsqrt%283%29%2Ai\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+1-sqrt%283%29%2Ai ALIGN=MIDDLE  ALT=\"z+=+1-sqrt%283%29%2Ai\"   DISABLED_style= \"max-width:90%; height:auto;\" >Break up the expression.  \r<BR>\n" );
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document.write( "So the next two solutions are are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+1%2Bsqrt%283%29%2Ai ALIGN=MIDDLE  ALT=\"z+=+1%2Bsqrt%283%29%2Ai\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+1-sqrt%283%29%2Ai ALIGN=MIDDLE  ALT=\"z+=+1-sqrt%283%29%2Ai\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "which approximate to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=1%2B1.732%2Ai ALIGN=MIDDLE  ALT=\"z=1%2B1.732%2Ai\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=1-1.732%2Ai ALIGN=MIDDLE  ALT=\"z=1-1.732%2Ai\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "-----------------------------------------------\r<BR>\n" );
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document.write( "Now let's solve  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E2%2B2%2Az%2B4=0 ALIGN=MIDDLE  ALT=\"z%5E2%2B2%2Az%2B4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E2%2B2z%2B4=0 ALIGN=MIDDLE  ALT=\"z%5E2%2B2z%2B4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation.\r<BR>\n" );
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document.write( "Notice we have a quadratic equation in the form of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=az%5E2%2Bbz%2Bc ALIGN=MIDDLE  ALT=\"az%5E2%2Bbz%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=2 ALIGN=MIDDLE  ALT=\"b=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=4 ALIGN=MIDDLE  ALT=\"c=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's use the quadratic formula to solve for z\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"z+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the quadratic formula\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-%282%29+%2B-+sqrt%28+%282%29%5E2-4%281%29%284%29+%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"z+=+%28-%282%29+%2B-+sqrt%28+%282%29%5E2-4%281%29%284%29+%29%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=2 ALIGN=MIDDLE  ALT=\"b=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=4 ALIGN=MIDDLE  ALT=\"c=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-2+%2B-+sqrt%28+4-4%281%29%284%29+%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"z+=+%28-2+%2B-+sqrt%28+4-4%281%29%284%29+%29%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-2+%2B-+sqrt%28+4-16+%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"z+=+%28-2+%2B-+sqrt%28+4-16+%29%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%281%29%284%29 ALIGN=MIDDLE  ALT=\"4%281%29%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16 ALIGN=MIDDLE  ALT=\"16\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-2+%2B-+sqrt%28+-12+%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"z+=+%28-2+%2B-+sqrt%28+-12+%29%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16 ALIGN=MIDDLE  ALT=\"16\"   DISABLED_style= \"max-width:90%; height:auto;\" > from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-12 ALIGN=MIDDLE  ALT=\"-12\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-2+%2B-+sqrt%28+-12+%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"z+=+%28-2+%2B-+sqrt%28+-12+%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-2+%2B-+2i%2Asqrt%283%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"z+=+%28-2+%2B-+2i%2Asqrt%283%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify the square root  (note: If you need help with simplifying square roots, check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)  \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-2%29%2F%282%29+%2B-+%282i%2Asqrt%283%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"z+=+%28-2%29%2F%282%29+%2B-+%282i%2Asqrt%283%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the fraction.  \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+-1+%2B-+sqrt%283%29%2Ai ALIGN=MIDDLE  ALT=\"z+=+-1+%2B-+sqrt%283%29%2Ai\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce.  \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+-1%2Bsqrt%283%29%2Ai ALIGN=MIDDLE  ALT=\"z+=+-1%2Bsqrt%283%29%2Ai\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+-1-sqrt%283%29%2Ai ALIGN=MIDDLE  ALT=\"z+=+-1-sqrt%283%29%2Ai\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the expression.  \r<BR>\n" );
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document.write( "So the next two solutions are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+-1%2Bsqrt%283%29%2Ai ALIGN=MIDDLE  ALT=\"z+=+-1%2Bsqrt%283%29%2Ai\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+-1-sqrt%283%29%2Ai ALIGN=MIDDLE  ALT=\"z+=+-1-sqrt%283%29%2Ai\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "which approximate to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=-1%2B1.732%2Ai ALIGN=MIDDLE  ALT=\"z=-1%2B1.732%2Ai\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=-1-1.732%2Ai ALIGN=MIDDLE  ALT=\"z=-1-1.732%2Ai\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "========================================================\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So altogether, we have the 6 solutions\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=-2 ALIGN=MIDDLE  ALT=\"z=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=2 ALIGN=MIDDLE  ALT=\"z=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+1%2Bsqrt%283%29%2Ai ALIGN=MIDDLE  ALT=\"z+=+1%2Bsqrt%283%29%2Ai\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+1-sqrt%283%29%2Ai ALIGN=MIDDLE  ALT=\"z+=+1-sqrt%283%29%2Ai\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+-1%2Bsqrt%283%29%2Ai ALIGN=MIDDLE  ALT=\"z+=+-1%2Bsqrt%283%29%2Ai\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+-1-sqrt%283%29%2Ai ALIGN=MIDDLE  ALT=\"z+=+-1-sqrt%283%29%2Ai\"   DISABLED_style= \"max-width:90%; height:auto;\" >  </SPAN>\n" );
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