document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1065005>Question 1065005</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%29x%5E2+-+sqrt%283%29x+%2B+k+=+0 ALIGN=MIDDLE  ALT=\"sqrt%282%29x%5E2+-+sqrt%283%29x+%2B+k+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > with k a constant has two solutions <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28theta%29 ALIGN=MIDDLE  ALT=\"sin%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28theta%29 ALIGN=MIDDLE  ALT=\"cos%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (0<=theta<=pi/2),then k = ...\r<BR>\n" );
document.write( "\n" );
document.write( "Hi, I got <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F4sqrt%282%29+%3E+k ALIGN=MIDDLE  ALT=\"3%2F4sqrt%282%29+%3E+k\"   DISABLED_style= \"max-width:90%; height:auto;\" >. What should I do next? Do I need to pick some lesser number than that? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #680143 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=680143\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I agree that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F4sqrt%282%29+%3E=+k ALIGN=MIDDLE  ALT=\"3%2F4sqrt%282%29+%3E=+k\"   DISABLED_style= \"max-width:90%; height:auto;\" > is necessary<BR>\n" );
document.write( "to get two real solutions, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d ALIGN=MIDDLE  ALT=\"d\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e ALIGN=MIDDLE  ALT=\"e\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "but we need more than that,<BR>\n" );
document.write( "and trying to use the quadratic formula seems awkward in this case.<BR>\n" );
document.write( "I would say that it is necessary and sufficient<BR>\n" );
document.write( "to have two real solutions, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d ALIGN=MIDDLE  ALT=\"d\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e ALIGN=MIDDLE  ALT=\"e\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "such that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28d%3E=0%2Ce%3E=0%2Cd%5E2%2Be%5E2=1%29 ALIGN=MIDDLE  ALT=\"system%28d%3E=0%2Ce%3E=0%2Cd%5E2%2Be%5E2=1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "For any quadratic equation, we know that<BR>\n" );
document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d ALIGN=MIDDLE  ALT=\"d\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e ALIGN=MIDDLE  ALT=\"e\"   DISABLED_style= \"max-width:90%; height:auto;\" > are solutions to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2+%2Bbx+%2B+c+=+0 ALIGN=MIDDLE  ALT=\"ax%5E2+%2Bbx+%2B+c+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28de=c%2Fa%2C%22and%22%2Cd%2Be=-b%2Fa%29%29 ALIGN=MIDDLE  ALT=\"system%28de=c%2Fa%2C%22and%22%2Cd%2Be=-b%2Fa%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "In this case,<BR>\n" );
document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d ALIGN=MIDDLE  ALT=\"d\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e ALIGN=MIDDLE  ALT=\"e\"   DISABLED_style= \"max-width:90%; height:auto;\" > are solutions to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%29x%5E2+-+sqrt%283%29x+%2B+k+=+0 ALIGN=MIDDLE  ALT=\"sqrt%282%29x%5E2+-+sqrt%283%29x+%2B+k+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28de=k%2Fsqrt%282%29%2C%22and%22%2Cd%2Be=sqrt%283%29%2Fsqrt%282%29%29 ALIGN=MIDDLE  ALT=\"system%28de=k%2Fsqrt%282%29%2C%22and%22%2Cd%2Be=sqrt%283%29%2Fsqrt%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "With <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%3E=0 ALIGN=MIDDLE  ALT=\"k%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , we would have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=de=k%2Fsqrt%282%29%3E0 ALIGN=MIDDLE  ALT=\"de=k%2Fsqrt%282%29%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , meaning <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%3E=0 ALIGN=MIDDLE  ALT=\"d%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%3E=0 ALIGN=MIDDLE  ALT=\"e%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Let us see if we can find a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%3E=0 ALIGN=MIDDLE  ALT=\"k%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > out of<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28d%5E2%2Be%5E2=1%2Cde=k%2Fsqrt%282%29%2Cd%2Be=sqrt%283%29%2Fsqrt%282%29%29 ALIGN=MIDDLE  ALT=\"system%28d%5E2%2Be%5E2=1%2Cde=k%2Fsqrt%282%29%2Cd%2Be=sqrt%283%29%2Fsqrt%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28d%5E2%2Be%5E2=1%2Cde=k%2Fsqrt%282%29%2Cd%2Be=sqrt%283%29%2Fsqrt%282%29%29 ALIGN=MIDDLE  ALT=\"system%28d%5E2%2Be%5E2=1%2Cde=k%2Fsqrt%282%29%2Cd%2Be=sqrt%283%29%2Fsqrt%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28d%5E2%2Be%5E2=1%2C2de=2k%2Fsqrt%282%29%2C%28d%2Be%29%5E2=3%2F2%29 ALIGN=MIDDLE  ALT=\"system%28d%5E2%2Be%5E2=1%2C2de=2k%2Fsqrt%282%29%2C%28d%2Be%29%5E2=3%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28d%5E2%2Be%5E2=1%2C2de=2k%2Fsqrt%282%29%2Cd%5E2%2Be%5E2%2B2de=3%2F2%29 ALIGN=MIDDLE  ALT=\"system%28d%5E2%2Be%5E2=1%2C2de=2k%2Fsqrt%282%29%2Cd%5E2%2Be%5E2%2B2de=3%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28d%5E2%2Be%5E2=1%2C2de=2k%2Fsqrt%282%29%2C1%2B2k%2Fsqrt%282%29=3%2F2%29 ALIGN=MIDDLE  ALT=\"system%28d%5E2%2Be%5E2=1%2C2de=2k%2Fsqrt%282%29%2C1%2B2k%2Fsqrt%282%29=3%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28d%5E2%2Be%5E2=1%2C2de=2k%2Fsqrt%282%29%2C2k%2Fsqrt%282%29=1%2F2%29 ALIGN=MIDDLE  ALT=\"system%28d%5E2%2Be%5E2=1%2C2de=2k%2Fsqrt%282%29%2C2k%2Fsqrt%282%29=1%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28d%5E2%2Be%5E2=1%2C2de=2k%2Fsqrt%282%29%2Chighlight%28k=sqrt%282%29%2F4%29%29 ALIGN=MIDDLE  ALT=\"system%28d%5E2%2Be%5E2=1%2C2de=2k%2Fsqrt%282%29%2Chighlight%28k=sqrt%282%29%2F4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "NOTE:<BR>\n" );
document.write( "The solutions to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%29x%5E2+-+sqrt%283%29x+%2B+k+=+0 ALIGN=MIDDLE  ALT=\"sqrt%282%29x%5E2+-+sqrt%283%29x+%2B+k+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28sqrt%283%29+%2B-+1%29%2F%282sqrt%282%29%29 ALIGN=MIDDLE  ALT=\"x=%28sqrt%283%29+%2B-+1%29%2F%282sqrt%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "which correspond to <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28theta=pi%2F12%2C%22or%22%2Ctheta=5pi%2F12%29 ALIGN=MIDDLE  ALT=\"system%28theta=pi%2F12%2C%22or%22%2Ctheta=5pi%2F12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );