document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=973666>Question 973666</A>: <I><SPAN STYLE=\"word-break: break-all;\"> what multipies to get -12 but adds to get -21? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #595814 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=595814\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> No pair of rational number factors will do that.<BR>\n" );
document.write( "The answer would be the solutions to the quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B21x-12=0 ALIGN=MIDDLE  ALT=\"x%5E2%2B21x-12=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "which are given by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-21+%2B-+sqrt%28489%29%29%2F2 ALIGN=MIDDLE  ALT=\"x=%28-21+%2B-+sqrt%28489%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That is approximately 0.55667219 and -21.55667219.<BR>\n" );
document.write( "Not all quadratic equations can be solved by factoring.<BR>\n" );
document.write( "(It only works if the solutions are rational numbers, and it is easiest when they are integers).<BR>\n" );
document.write( "However, there are two ways that work well to solve any quadratic equation.<BR>\n" );
document.write( " <BR>\n" );
document.write( "You can solve any quadratic equation of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > by using the quadratic formula:<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>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B21x-12=0 ALIGN=MIDDLE  ALT=\"x%5E2%2B21x-12=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28a=1%2Cb=21%2Cc=-12%29 ALIGN=MIDDLE  ALT=\"system%28a=1%2Cb=21%2Cc=-12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-21+%2B-+sqrt%2821%5E2-4%2A1%2A%28-12%29%29%29%2F%282%2A1%29=+%28-21+%2B-+sqrt%28441%2B48%29%29%2F2=+%28-21+%2B-+sqrt%28489%29%29%2F2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "Also, you can solve any quadratic equation by completing the square\",<BR>\n" );
document.write( "without having to remember the quadratic formula.<BR>\n" );
document.write( "(In some cases, it is easier than using the quadratic formula,<BR>\n" );
document.write( "but this is not one of those cases).<BR>\n" );
document.write( "In this case, you would do it like this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B21x-12=0 ALIGN=MIDDLE  ALT=\"x%5E2%2B21x-12=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%2B21x=12 ALIGN=MIDDLE  ALT=\"x%5E2%2B21x=12\"   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%2B21x%2B%2821%2F2%29%5E2=12%2B%2821%2F2%29%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2B21x%2B%2821%2F2%29%5E2=12%2B%2821%2F2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B21%2F2%29%5E2=12%2B441%2F4 ALIGN=MIDDLE  ALT=\"%28x%2B21%2F2%29%5E2=12%2B441%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B21%2F2%29%5E2=48%2F4%2B441%2F4 ALIGN=MIDDLE  ALT=\"%28x%2B21%2F2%29%5E2=48%2F4%2B441%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B21%2F2%29%5E2=489%2F4 ALIGN=MIDDLE  ALT=\"%28x%2B21%2F2%29%5E2=489%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x%2B21=sqrt%28489%2F4%29=sqrt%28489%29%2F2%2C%22or%22%2Cx%2B21=-sqrt%28489%2F4%29=-sqrt%28489%29%2F2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=-21%2Bsqrt%28489%2F4%29=-21%2Bsqrt%28489%29%2F2%2C%22or%22%2Cx=-21-sqrt%28489%2F4%29=-21-sqrt%28489%29%2F2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > <---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-21+%2B-+sqrt%28489%29%29%2F2 ALIGN=MIDDLE  ALT=\"x=%28-21+%2B-+sqrt%28489%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );