document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=184671>Question 184671</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For what value(s)of k will the graph of y=3x^2+(k+2)x+3 touch the x-axis at exactly one point </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #138606 by </B><A HREF=/tutors/aboutme.mpl?userid=josmiceli><B>josmiceli(19441)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josmiceli><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=josmiceli><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=138606\">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=y+=+3x%5E2+%2B+%28k%2B2%29x+%2B+3 ALIGN=MIDDLE  ALT=\"y+=+3x%5E2+%2B+%28k%2B2%29x+%2B+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Instead of 2 roots, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5B1%5D ALIGN=MIDDLE  ALT=\"r%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5B2%5D ALIGN=MIDDLE  ALT=\"r%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "There will be a double root, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x+-+r%29%28x+-+r%29+=+0 ALIGN=MIDDLE  ALT=\"%28x+-+r%29%28x+-+r%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+2rx+%2B+r%5E2+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+-+2rx+%2B+r%5E2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-------------------------<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2+%2B+%28k%2B2%29x+%2B+3+=+0 ALIGN=MIDDLE  ALT=\"3x%5E2+%2B+%28k%2B2%29x+%2B+3+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+%28%28k%2B2%29%2F3%29%2Ax+%2B+1+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+%28%28k%2B2%29%2F3%29%2Ax+%2B+1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "I can make the substitutions<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28k%2B2%29%2F3+=+-2r ALIGN=MIDDLE  ALT=\"%28k%2B2%29%2F3+=+-2r\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5E2+=+1 ALIGN=MIDDLE  ALT=\"r%5E2+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+1 ALIGN=MIDDLE  ALT=\"r+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+-1 ALIGN=MIDDLE  ALT=\"r+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-------------------------<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28k%2B2%29%2F3+=+-2%2A1 ALIGN=MIDDLE  ALT=\"%28k%2B2%29%2F3+=+-2%2A1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+%2B+2+=+-6 ALIGN=MIDDLE  ALT=\"k+%2B+2+=+-6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+-8 ALIGN=MIDDLE  ALT=\"k+=+-8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28k%2B2%29%2F3+=+-2%2A%28-1%29 ALIGN=MIDDLE  ALT=\"%28k%2B2%29%2F3+=+-2%2A%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+%2B+2+=+6 ALIGN=MIDDLE  ALT=\"k+%2B+2+=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+4 ALIGN=MIDDLE  ALT=\"k+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The graph touches the x-axis when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+-8 ALIGN=MIDDLE  ALT=\"k+=+-8\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+4 ALIGN=MIDDLE  ALT=\"k+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+-8 ALIGN=MIDDLE  ALT=\"k+=+-8\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+3x%5E2+%2B+%28k%2B2%29x+%2B+3 ALIGN=MIDDLE  ALT=\"y+=+3x%5E2+%2B+%28k%2B2%29x+%2B+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+3x%5E2+%2B+%28-8%2B2%29x+%2B+3 ALIGN=MIDDLE  ALT=\"y+=+3x%5E2+%2B+%28-8%2B2%29x+%2B+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+3x%5E2+-+6x+%2B+3 ALIGN=MIDDLE  ALT=\"y+=+3x%5E2+-+6x+%2B+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+4 ALIGN=MIDDLE  ALT=\"k+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+3x%5E2+%2B+%284%2B2%29x+%2B+3 ALIGN=MIDDLE  ALT=\"y+=+3x%5E2+%2B+%284%2B2%29x+%2B+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+3x%5E2+%2B+6x+%2B+3 ALIGN=MIDDLE  ALT=\"y+=+3x%5E2+%2B+6x+%2B+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "I'll plot both:<BR>\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+500%2C+500%2C+-5%2C+5%2C+-2%2C+8%2C+3x%5E2+-+6x+%2B+3%2C3x%5E2+%2B+6x+%2B+3%29+ ALIGN=MIDDLE  ALT=\"+graph%28+500%2C+500%2C+-5%2C+5%2C+-2%2C+8%2C+3x%5E2+-+6x+%2B+3%2C3x%5E2+%2B+6x+%2B+3%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );