document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=290119>Question 290119</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve by using the quadratic formula. <BR>\n" );
document.write( "The solution set is { , } or is solution set 0?\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+3p%5E2=-13p-12+ ALIGN=MIDDLE  ALT=\"+3p%5E2=-13p-12+\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #210028 by </B><A HREF=/tutors/aboutme.mpl?userid=Deina><B>Deina(147)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Deina><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=Deina><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=210028\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First let's convert <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+3p%5E2+=+-13p+-12+ ALIGN=MIDDLE  ALT=\"+3p%5E2+=+-13p+-12+\"   DISABLED_style= \"max-width:90%; height:auto;\" > to a quadratic, <BR>\n" );
document.write( "in the form of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+ax%5E2+%2B+bx+%2B+c+=+0+ ALIGN=MIDDLE  ALT=\"+ax%5E2+%2B+bx+%2B+c+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" > so:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+3p%5E2+=+highlight%28-13p+-12%29+ ALIGN=MIDDLE  ALT=\"+3p%5E2+=+highlight%28-13p+-12%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > becomes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+3p%5E2+%2B+highlight%28+13p+%2B+12%29+=+0+ ALIGN=MIDDLE  ALT=\"+3p%5E2+%2B+highlight%28+13p+%2B+12%29+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <br><BR>\n" );
document.write( "Now plug those numbers into the quadratic formula, <BR>\n" );
document.write( "replacing a b & c as above:<BR>\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "a = 3<BR>\n" );
document.write( "b = 13<BR>\n" );
document.write( "c = 12<BR>\n" );
document.write( "Replacing \"x\" with \"p\" <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x%29+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"highlight%28x%29+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Becomes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28p%29+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"highlight%28p%29+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br>\r<BR>\n" );
document.write( "\n" );
document.write( "Replacing \"a\" with \"3\" <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Ahighlight%28a%29%2Ac+%29%29%2F%282%2Ahighlight%28a%29%29+ ALIGN=MIDDLE  ALT=\"p+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Ahighlight%28a%29%2Ac+%29%29%2F%282%2Ahighlight%28a%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > becomes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Ahighlight%283%29%2Ac+%29%29%2F%282%2Ahighlight%283%29%29+ ALIGN=MIDDLE  ALT=\"p+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Ahighlight%283%29%2Ac+%29%29%2F%282%2Ahighlight%283%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br>\r<BR>\n" );
document.write( "\n" );
document.write( "Replacing \"b\" with \"13\" <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p+=+%28-highlight%28b%29+%2B-+sqrt%28+highlight%28b%29%5E2-4%2A3%2Ac+%29%29%2F%282%2A3%29+ ALIGN=MIDDLE  ALT=\"p+=+%28-highlight%28b%29+%2B-+sqrt%28+highlight%28b%29%5E2-4%2A3%2Ac+%29%29%2F%282%2A3%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > becomes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p+=+%28-highlight%2813%29+%2B-+sqrt%28+highlight%2813%29%5E2-4%2A3%2Ac+%29%29%2F%282%2A3%29+ ALIGN=MIDDLE  ALT=\"p+=+%28-highlight%2813%29+%2B-+sqrt%28+highlight%2813%29%5E2-4%2A3%2Ac+%29%29%2F%282%2A3%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br>\r<BR>\n" );
document.write( "\n" );
document.write( "Replacing \"c\" with \"12\" <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p+=+%28-13+%2B-+sqrt%28+13%5E2-4%2A3%2Ahighlight%28c%29+%29%29%2F%282%2A3%29+ ALIGN=MIDDLE  ALT=\"p+=+%28-13+%2B-+sqrt%28+13%5E2-4%2A3%2Ahighlight%28c%29+%29%29%2F%282%2A3%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > becomes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p+=+%28-13+%2B-+sqrt%28+13%5E2-4%2A3%2Ahighlight%2812%29+%29%29%2F%282%2A3%29+ ALIGN=MIDDLE  ALT=\"p+=+%28-13+%2B-+sqrt%28+13%5E2-4%2A3%2Ahighlight%2812%29+%29%29%2F%282%2A3%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Mr. Ichudov's Quadratic Solver explains the remaining process better than I:<BR>\n" );
document.write( "<TABLE  BORDER=1>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=quadratic.solver>SOLVE quadratic equation with variable</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ap%5E2%2Bbp%2Bc=0 ALIGN=MIDDLE  ALT=\"ap%5E2%2Bbp%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3p%5E2%2B13p%2B12+=+0 ALIGN=MIDDLE  ALT=\"3p%5E2%2B13p%2B12+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) has the following solutons:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"p%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For these solutions to exist, the <U>discriminant</U> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > should not be a negative number.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First, we need to compute the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=%2813%29%5E2-4%2A3%2A12=25 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%2813%29%5E2-4%2A3%2A12=25\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=25 is greater than zero. That means that there are two solutions: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B12%5D+=+%28-13%2B-sqrt%28+25+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28-13%2B-sqrt%28+25+%29%29%2F2%5Ca\"   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=p%5B1%5D+=+%28-%2813%29%2Bsqrt%28+25+%29%29%2F2%5C3+=+-1.33333333333333 ALIGN=MIDDLE  ALT=\"p%5B1%5D+=+%28-%2813%29%2Bsqrt%28+25+%29%29%2F2%5C3+=+-1.33333333333333\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%5B2%5D+=+%28-%2813%29-sqrt%28+25+%29%29%2F2%5C3+=+-3 ALIGN=MIDDLE  ALT=\"p%5B2%5D+=+%28-%2813%29-sqrt%28+25+%29%29%2F2%5C3+=+-3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <BR>\n" );
document.write( "    Quadratic expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3p%5E2%2B13p%2B12 ALIGN=MIDDLE  ALT=\"3p%5E2%2B13p%2B12\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be factored:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3p%5E2%2B13p%2B12+=+3%28p--1.33333333333333%29%2A%28p--3%29 ALIGN=MIDDLE  ALT=\"3p%5E2%2B13p%2B12+=+3%28p--1.33333333333333%29%2A%28p--3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: -1.33333333333333, -3.\n" );
document.write( "Here's your graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B13%2Ax%2B12+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B13%2Ax%2B12+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "\n" );
document.write( "So your solutions are: p=-1.33333333333333 and p=-3. and bob's your uncle! </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );