document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=767452>Question 767452</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Dear Sir/Madam,<BR>\n" );
document.write( "Good day! Kindly help me on this problem:<BR>\n" );
document.write( "Find the value of k so that the equation 3x^2-4kx+k = 0 will have real roots. Thank you very much for your help! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #467628 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=467628\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "Your question cannot be answered as you asked it.  You cannot find <i><b>the</b></i> value of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE k\"> such that the equation <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 3x^2\ -\ 4kx\ +\ k\ =\ 0\"> has real roots.  That is because the set of values of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE k\"> that fulfills the conditions of the problem has infinite elements.\r<BR>\n" );
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document.write( "In order for a quadratic equation, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE ax^2\ +\ bx\ +\ c\ =\ 0\"> to have real roots, the discriminant, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE b^2\ -\ 4ac\"> must be non-negative.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (4k)^2\ -\ (4)(3)(k)\ \geq\ 0\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 4k(4k\ -\ 3)\ \geq\ 0\">\r<BR>\n" );
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document.write( "Critical points are <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE k\ =\ 0\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE k\ =\ \frac{3}{4}\">\r<BR>\n" );
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document.write( "Test value:  -1:  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ 4(-1)(4(-1)\ -\ 3)\ =\ 28\ >\ 0\">\r<BR>\n" );
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document.write( "Test value:  0.5: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ 4(.5)(4(.5)\ -\ 3)\ =\ -2\ <\ 0\">\r<BR>\n" );
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document.write( "Test value:  1:  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ 4(1)(4(1)\ -\ 3)\ =\ 4\ >\ 0\">\r<BR>\n" );
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document.write( "So the intervals where values of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE k\"> result in real roots for the given quadratic are:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ -\infty\ <\ k\ \leq\ 0\">\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/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{3}{4}\ \leq\ k\ <\ \infty\">\r<BR>\n" );
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document.write( "And a compact description of the solution set is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left\{k\ \in\ \mathbb{R}\ |\ k\ \not\in\ \left(0,\frac{3}{4}\right)\right\}\"> \r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "<font face=\"Math1\" size=\"+2\">Egw to Beta kai to Sigma</font><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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