document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1050486>Question 1050486</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the range of values of p for which 4x^2+12x+15=p(4x+5) 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 #666035 by </B><A HREF=/tutors/aboutme.mpl?userid=Fombitz><B>Fombitz(32388)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Fombitz><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=Fombitz><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=666035\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Use the discriminant,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2%2B12x%2B15=4px%2B5p ALIGN=MIDDLE  ALT=\"4x%5E2%2B12x%2B15=4px%2B5p\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2%2B12x-4px%2B15-5p=0 ALIGN=MIDDLE  ALT=\"4x%5E2%2B12x-4px%2B15-5p=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2%2B%2812-4p%29x%2B%2815-5p%29=0 ALIGN=MIDDLE  ALT=\"4x%5E2%2B%2812-4p%29x%2B%2815-5p%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So then,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=b%5E2-4ac ALIGN=MIDDLE  ALT=\"D=b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=%2812-4p%29%5E2-4%2815-5p%29%284%29 ALIGN=MIDDLE  ALT=\"D=%2812-4p%29%5E2-4%2815-5p%29%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For real roots,<BR>\n" );
document.write( "<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;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%2812-4p%29%5E2-16%2815-5p%29%3E=0+ ALIGN=MIDDLE  ALT=\"+%2812-4p%29%5E2-16%2815-5p%29%3E=0+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28144-96p%2B16p%5E2%29-240%2B80p%3E=0 ALIGN=MIDDLE  ALT=\"%28144-96p%2B16p%5E2%29-240%2B80p%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16p%5E2-16p-96%3E0 ALIGN=MIDDLE  ALT=\"16p%5E2-16p-96%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%5E2-p-6%3E=0 ALIGN=MIDDLE  ALT=\"p%5E2-p-6%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28p-3%29%28p%2B2%29%3E=0 ALIGN=MIDDLE  ALT=\"%28p-3%29%28p%2B2%29%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Use the critical points, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p=3 ALIGN=MIDDLE  ALT=\"p=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p=-2 ALIGN=MIDDLE  ALT=\"p=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "Break up the number line into 3 regions,<BR>\n" );
document.write( "Region 1 : <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%3C-2 ALIGN=MIDDLE  ALT=\"p%3C-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Region 2 : <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%3Cp%3C3 ALIGN=MIDDLE  ALT=\"-2%3Cp%3C3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Region 3 : <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%3E3 ALIGN=MIDDLE  ALT=\"p%3E3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Choose a value in each region and check the inequality,<BR>\n" );
document.write( "Region 1 : <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p=-3 ALIGN=MIDDLE  ALT=\"p=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-3-3%29%28-3%2B2%29%3E=0 ALIGN=MIDDLE  ALT=\"%28-3-3%29%28-3%2B2%29%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6%28-1%29%3E=0 ALIGN=MIDDLE  ALT=\"-6%28-1%29%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%3E=0 ALIGN=MIDDLE  ALT=\"6%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "True<BR>\n" );
document.write( "Region 2 : <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p=0 ALIGN=MIDDLE  ALT=\"p=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%280-3%29%280%2B2%29%3E=0 ALIGN=MIDDLE  ALT=\"%280-3%29%280%2B2%29%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6%3E=0 ALIGN=MIDDLE  ALT=\"-6%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "False<BR>\n" );
document.write( "Region 3 : <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p=4 ALIGN=MIDDLE  ALT=\"p=4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284-3%29%284%2B2%29%3E=0 ALIGN=MIDDLE  ALT=\"%284-3%29%284%2B2%29%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%286%29%3E=0 ALIGN=MIDDLE  ALT=\"1%286%29%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%3E=0 ALIGN=MIDDLE  ALT=\"6%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "True<BR>\n" );
document.write( "So then,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%3C=-2 ALIGN=MIDDLE  ALT=\"p%3C=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >U<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%3E=3 ALIGN=MIDDLE  ALT=\"p%3E=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );