document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=948573>Question 948573</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me answer this: For what values of k will the equation 3x^2=k-2x have 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 #579007 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=579007\">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=3x%5E2=k-2x ALIGN=MIDDLE  ALT=\"3x%5E2=k-2x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2B2x-k=0 ALIGN=MIDDLE  ALT=\"3x%5E2%2B2x-k=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >=> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=3 ALIGN=MIDDLE  ALT=\"a=3\"   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=-k ALIGN=MIDDLE  ALT=\"c=-k\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The argument of the square root, the expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2+%96+4ac ALIGN=MIDDLE  ALT=\"b%5E2+%96+4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >, is called the \"discriminant\" because, by using its value, you can discriminate between (tell the differences between) the various solution types.\r<BR>\n" );
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document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2+%96+4ac%3C0 ALIGN=MIDDLE  ALT=\"b%5E2+%96+4ac%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > there is no real roots<BR>\n" );
document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2+%96+4ac=0 ALIGN=MIDDLE  ALT=\"b%5E2+%96+4ac=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > there is one real roots<BR>\n" );
document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2+%96+4ac%3E0 ALIGN=MIDDLE  ALT=\"b%5E2+%96+4ac%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > there are two real roots\r<BR>\n" );
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document.write( "so, we need<BR>\n" );
document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2+%96+4ac=0 ALIGN=MIDDLE  ALT=\"b%5E2+%96+4ac=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > there is one real roots<BR>\n" );
document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2+%96+4ac%3E0 ALIGN=MIDDLE  ALT=\"b%5E2+%96+4ac%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > there are two real roots\r<BR>\n" );
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document.write( "and I will write both in one as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2+%96+4ac%3E=0 ALIGN=MIDDLE  ALT=\"b%5E2+%96+4ac%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "now just plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=3 ALIGN=MIDDLE  ALT=\"a=3\"   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=-k ALIGN=MIDDLE  ALT=\"c=-k\"   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=2%5E2+%96+4%2A3%2A%28-k%29%3E=0 ALIGN=MIDDLE  ALT=\"2%5E2+%96+4%2A3%2A%28-k%29%3E=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+%96+12%28-k%29%3E=0 ALIGN=MIDDLE  ALT=\"4+%96+12%28-k%29%3E=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+%2B12k%3E=0 ALIGN=MIDDLE  ALT=\"4+%2B12k%3E=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=12k%3E=-4 ALIGN=MIDDLE  ALT=\"12k%3E=-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=k%3E=-4%2F12 ALIGN=MIDDLE  ALT=\"k%3E=-4%2F12\"   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=k%3E=-1%2F3 ALIGN=MIDDLE  ALT=\"k%3E=-1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "so, if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-1%2F3 ALIGN=MIDDLE  ALT=\"k=-1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > we will have one real root which is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1%2F3 ALIGN=MIDDLE  ALT=\"x=-1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > (you can check it)<BR>\n" );
document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%3E=-1%2F3 ALIGN=MIDDLE  ALT=\"k%3E=-1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, using for example <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=0 ALIGN=MIDDLE  ALT=\"k=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > we will have two real roots which are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+0 ALIGN=MIDDLE  ALT=\"x+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-2%2F3 ALIGN=MIDDLE  ALT=\"x=-2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "so, to get real root/roots, you can use any value for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" > from [<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F3 ALIGN=MIDDLE  ALT=\"-1%2F3\"   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( "  </SPAN>\n" );
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