document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=990252>Question 990252</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve<BR>\n" );
document.write( "(x^2-x)^2+(x^2-x)-2=0 in exact values </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #610287 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52781)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=610287\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "Your equation is\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2-x%29%5E2%2B%28x%5E2-x%29-2 ALIGN=MIDDLE  ALT=\"%28x%5E2-x%29%5E2%2B%28x%5E2-x%29-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "The standard and universal method for solving such equations is introducing new variable.\r<BR>\n" );
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document.write( "In our case let &nbsp;y = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-x ALIGN=MIDDLE  ALT=\"x%5E2-x\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Then the original equation takes the form\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2+%2B+y+-+2 ALIGN=MIDDLE  ALT=\"y%5E2+%2B+y+-+2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "It is a quadratic equation. &nbsp;You can easily solve it by applying the quadratic formula or the Viete's theorem. &nbsp;Its roots are \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D ALIGN=MIDDLE  ALT=\"y%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > = -2, &nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B2%5D ALIGN=MIDDLE  ALT=\"y%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 1.\r<BR>\n" );
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document.write( "Now we should solve two quadratic equations to find &nbsp;<B>x</B>. &nbsp;They are\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+x ALIGN=MIDDLE  ALT=\"x%5E2+-+x\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2 ALIGN=MIDDLE  ALT=\"-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(1) \r<BR>\n" );
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document.write( "and\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-x ALIGN=MIDDLE  ALT=\"x%5E2+-x\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(2)\r<BR>\n" );
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document.write( "The first of these two equations is \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+x+%2B+2 ALIGN=MIDDLE  ALT=\"x%5E2+-+x+%2B+2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "It has no real solutions &nbsp;(the discriminant &nbsp;d = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 1 - 4*2 = -7 &nbsp;is negative).\r<BR>\n" );
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document.write( "The second of these two equations is\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-x-1 ALIGN=MIDDLE  ALT=\"x%5E2-x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "It has two roots &nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%2C2%5D ALIGN=MIDDLE  ALT=\"x%5B1%2C2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281+%2B-+sqrt%285%29%29%2F2 ALIGN=MIDDLE  ALT=\"%281+%2B-+sqrt%285%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "<B>Answer</B>. &nbsp;The given equation has two roots: &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Bsqrt%285%29%29%2F2 ALIGN=MIDDLE  ALT=\"%281%2Bsqrt%285%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > &nbsp;and &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281-sqrt%285%29%29%2F2 ALIGN=MIDDLE  ALT=\"%281-sqrt%285%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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