document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1132053>Question 1132053</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If &#945; and &#946; are the roots of 2x²-2x+5=0, find the values of (1/&#945;+1)+(1/&#946;).<BR>\n" );
document.write( "Hence determine the nature of the roots.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #748923 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20849)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=748923\">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=2x%5E2-2x%2B5=0+ ALIGN=MIDDLE  ALT=\"2x%5E2-2x%2B5=0+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "we need to use  quadratic formula to find roots<BR>\n" );
document.write( "and, to determine the nature of the roots we can check if discriminant is  <<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( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2+-+4ac+%3C+0+ ALIGN=MIDDLE  ALT=\"b%5E2+-+4ac+%3C+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" > then, as we know,  the roots are<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+imaginary+ ALIGN=MIDDLE  ALT=\"+imaginary+\"   DISABLED_style= \"max-width:90%; height:auto;\" >values\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-2%29%5E2+-+4%2A2%2A5+%3C+0+ ALIGN=MIDDLE  ALT=\"%28-2%29%5E2+-+4%2A2%2A5+%3C+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=4+-+40+%3C+0+ ALIGN=MIDDLE  ALT=\"4+-+40+%3C+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=-+36+%3C+0+ ALIGN=MIDDLE  ALT=\"-+36+%3C+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" >=> the roots are imaginary \r<BR>\n" );
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document.write( "now use  quadratic formula to find roots\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"   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=x+=+%28-%28-2%29+%2B-+sqrt%28+%28-2%29%5E2-4%2A2%2A5+%29%29%2F%282%2A2%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-%28-2%29+%2B-+sqrt%28+%28-2%29%5E2-4%2A2%2A5+%29%29%2F%282%2A2%29+\"   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=x+=+%282+%2B-+sqrt%28+4-36%29%29%2F4+ ALIGN=MIDDLE  ALT=\"x+=+%282+%2B-+sqrt%28+4-36%29%29%2F4+\"   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=x+=+%282+%2B-+sqrt%28+-32%29%29%2F4+ ALIGN=MIDDLE  ALT=\"x+=+%282+%2B-+sqrt%28+-32%29%29%2F4+\"   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=x+=+%282+%2B-+sqrt%28+-2%2A16%29%29%2F4+ ALIGN=MIDDLE  ALT=\"x+=+%282+%2B-+sqrt%28+-2%2A16%29%29%2F4+\"   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=x+=+%282+%2B-+4sqrt%282%29%2Ai%29%2F4+ ALIGN=MIDDLE  ALT=\"x+=+%282+%2B-+4sqrt%282%29%2Ai%29%2F4+\"   DISABLED_style= \"max-width:90%; height:auto;\" >...simplify\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%281+%2B-+2sqrt%282%29%2Ai%29%2F2+ ALIGN=MIDDLE  ALT=\"x+=+%281+%2B-+2sqrt%282%29%2Ai%29%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "roots:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%281+%2B2sqrt%282%29%2Ai%29%2F2+ ALIGN=MIDDLE  ALT=\"x+=+%281+%2B2sqrt%282%29%2Ai%29%2F2+\"   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=x+=+%281+-+2sqrt%282%29%2Ai%29%2F2+ ALIGN=MIDDLE  ALT=\"x+=+%281+-+2sqrt%282%29%2Ai%29%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha=+%281+%2B2sqrt%282%29%2Ai%29%2F2+ ALIGN=MIDDLE  ALT=\"alpha=+%281+%2B2sqrt%282%29%2Ai%29%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=beta=%281+-+2sqrt%282%29%2Ai%29%2F2+ ALIGN=MIDDLE  ALT=\"beta=%281+-+2sqrt%282%29%2Ai%29%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "then,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%28alpha%2B1%29%2B%281%2Fbeta%29=1%2F%28%281+%2B2sqrt%282%29%2Ai%29%2F2+%2B1%29%2B%281%2F%281+-+2sqrt%282%29%2Ai%29%2F2+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%28alpha%2B1%29%2B%281%2Fbeta%29=1%2F%28%281+%2B2sqrt%282%29%2Ai%2B2%29%2F2+%29%2B%282%2F%281+-+2sqrt%282%29%2Ai+%29%29 ALIGN=MIDDLE   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=1%2F%28alpha%2B1%29%2B%281%2Fbeta%29=2%2F%283%2B2sqrt%282%29%2Ai%29%2B%282%2F%281+-+2sqrt%282%29%2Ai+%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );