document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=846997>Question 846997</A>: <I><SPAN STYLE=\"word-break: break-all;\"> X^4+x^2+1=0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #510184 by </B><A HREF=/tutors/aboutme.mpl?userid=josh_jordan><B>josh_jordan(263)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josh_jordan><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=josh_jordan><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=510184\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is quite the complex problem!  So, let's get started.\r<BR>\n" );
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document.write( "If we rewrite <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4%2Bx%5E2%2B1=0 ALIGN=MIDDLE  ALT=\"x%5E4%2Bx%5E2%2B1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > in the form of a quadratic equation, we will be able to find our zeroes.  We can do this by rewriting as:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%29%5E2%2B%28x%29x%2B1=0 ALIGN=MIDDLE  ALT=\"%28x%5E2%29%5E2%2B%28x%29x%2B1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This puts our equation in standard quadratic form of ax^2+bx+c=0.  Now, we can replace the first x inside the parenthesis with a letter and replace the second x on the outside of the parenthesis with the same letter.  Let's use u:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u%5E2%2Bu%2B1=0 ALIGN=MIDDLE  ALT=\"u%5E2%2Bu%2B1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now we can solve using the quadratic formula.  This will give us:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u=%28-1+%2B-+sqrt%28+-3+%29%29%2F%282%29+ ALIGN=MIDDLE  ALT=\"u=%28-1+%2B-+sqrt%28+-3+%29%29%2F%282%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now, here comes the tricky part.  We have to replace our u with our original variable and power:  x^2, which will turn this into:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=%28-1+%2B-+sqrt%28+-3+%29%29%2F%282%29+ ALIGN=MIDDLE  ALT=\"x%5E2=%28-1+%2B-+sqrt%28+-3+%29%29%2F%282%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Because we want the value of x and not x^2, we will have to take the square root of both sides.  Remember, when we take the square root of both sides of an equation, we have to add the +- sign in front.  Doing this is going to give us 4 zeros:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=+%2B-+%28sqrt%28%28-1+%2B-+sqrt%28+-3+%29%29%2F%282%29%29%29 ALIGN=MIDDLE  ALT=\"x=+%2B-+%28sqrt%28%28-1+%2B-+sqrt%28+-3+%29%29%2F%282%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Next, you will want to convert the square root of -3 into an imaginary:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=+%2B-+%28sqrt%28%28-1+%2B-+sqrt%28+3+%29%2Ai%29%2F%282%29%29%29 ALIGN=MIDDLE  ALT=\"x=+%2B-+%28sqrt%28%28-1+%2B-+sqrt%28+3+%29%2Ai%29%2F%282%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Finally, you will need to rationalize the denominator since the denominator of the fraction is the square root of 2.  This will give you your final answer:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=+%2B-+%28sqrt%28%282%28-1+%2B-+sqrt%28+3+%29%2Ai%29%29%29%2F%282%29%29 ALIGN=MIDDLE  ALT=\"x=+%2B-+%28sqrt%28%282%28-1+%2B-+sqrt%28+3+%29%2Ai%29%29%29%2F%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "For some reason, the answer typed out the word \"PLUS_MINNUS-\" instead of \"+-\".  Just replace the \"PLUS_MINNUS-\" with the plus/minus symbol (+-).\r<BR>\n" );
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document.write( "NOTE:  You will notice the numerator of the fraction contains a square root inside of a square root (nested root).  You will more than likely not be required to denest this root, so it was not denested in this solution. </SPAN>\n" );
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