document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1129569>Question 1129569</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the range of values for which the equation x^3-qx+q+3=0 has real roots </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #746134 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</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=746134\">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%5E3-qx%2Bq%2B3=0+ ALIGN=MIDDLE  ALT=\"x%5E3-qx%2Bq%2B3=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=x%5E3-q%28x-1-3%29=0+ ALIGN=MIDDLE  ALT=\"x%5E3-q%28x-1-3%29=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=x%5E3=q%28x-4%29+ ALIGN=MIDDLE  ALT=\"x%5E3=q%28x-4%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=q=x%5E3%2F%28x-4%29+ ALIGN=MIDDLE  ALT=\"q=x%5E3%2F%28x-4%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > -><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C%3E4 ALIGN=MIDDLE  ALT=\"x%3C%3E4\"   DISABLED_style= \"max-width:90%; height:auto;\" >; => to have real roots, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > must be greater than <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( "the range of values:\r<BR>\n" );
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document.write( " (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=infinity ALIGN=MIDDLE  ALT=\"infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "check few of them:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=4.1 ALIGN=MIDDLE  ALT=\"x=4.1\"   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=q=4.1%5E3%2F%284.1-4%29=689.21 ALIGN=MIDDLE  ALT=\"q=4.1%5E3%2F%284.1-4%29=689.21\"   DISABLED_style= \"max-width:90%; height:auto;\" >=><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-689.21x%2B689.21%2B3=0 ALIGN=MIDDLE  ALT=\"x%5E3-689.21x%2B689.21%2B3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >=><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-689.21x-692.21=0 ALIGN=MIDDLE  ALT=\"x%5E3-689.21x-692.21=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=+graph%28+600%2C+600%2C+-50%2C+50%2C+-5000%2C+5000%2Cx%5E3-689.21x-692.21%29+ ALIGN=MIDDLE  ALT=\"+graph%28+600%2C+600%2C+-50%2C+50%2C+-5000%2C+5000%2Cx%5E3-689.21x-692.21%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=5 ALIGN=MIDDLE  ALT=\"x=5\"   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=q=5%5E3%2F%285-4%29=75 ALIGN=MIDDLE  ALT=\"q=5%5E3%2F%285-4%29=75\"   DISABLED_style= \"max-width:90%; height:auto;\" >=><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-75x%2B75%2B3=0 ALIGN=MIDDLE  ALT=\"x%5E3-75x%2B75%2B3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >=><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-75x-73=0 ALIGN=MIDDLE  ALT=\"x%5E3-75x-73=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=+graph%28+600%2C+600%2C+-30%2C+30%2C+-300%2C+300%2Cx%5E3-75x-73%29+ ALIGN=MIDDLE  ALT=\"+graph%28+600%2C+600%2C+-30%2C+30%2C+-300%2C+300%2Cx%5E3-75x-73%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=5 ALIGN=MIDDLE  ALT=\"x=5\"   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=q=6%5E3%2F%286-4%29=108 ALIGN=MIDDLE  ALT=\"q=6%5E3%2F%286-4%29=108\"   DISABLED_style= \"max-width:90%; height:auto;\" >=><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-108x%2B108%2B3=0 ALIGN=MIDDLE  ALT=\"x%5E3-108x%2B108%2B3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >=><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-108x-111=0 ALIGN=MIDDLE  ALT=\"x%5E3-108x-111=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=+graph%28+600%2C+600%2C+-30%2C+30%2C+-300%2C+300%2Cx%5E3-108x-111%29+ ALIGN=MIDDLE  ALT=\"+graph%28+600%2C+600%2C+-30%2C+30%2C+-300%2C+300%2Cx%5E3-108x-111%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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