document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=166538>Question 166538</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the sum of the roots of the equation <BR>\n" );
document.write( "ax^2+bx+c=0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #122733 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=122733\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Remember the quadratic formula is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > note: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3C%3E0 ALIGN=MIDDLE  ALT=\"a%3C%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w=sqrt%28+b%5E2-4ac+%29 ALIGN=MIDDLE  ALT=\"w=sqrt%28+b%5E2-4ac+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (to simplify things a bit)\r<BR>\n" );
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document.write( "So the quadratic formula becomes\r<BR>\n" );
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document.write( "which really breaks down to\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b+%2B+w%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"x+=+%28-b+%2B+w%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b+-+w%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"x+=+%28-b+-+w%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the first root is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D+=+%28-b+%2B+w%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-b+%2B+w%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the second root is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D+=+%28-b+-+w%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-b+-+w%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "==========================================\r<BR>\n" );
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document.write( "<h4>Adding the Roots:</h4>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D%2Bx%5B2%5D ALIGN=MIDDLE  ALT=\"x%5B1%5D%2Bx%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now let's add the roots\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-b+%2B+w%29%2F%282a%29%2B%28-b+-+w%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"%28-b+%2B+w%29%2F%282a%29%2B%28-b+-+w%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D+=+%28-b+%2B+w%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-b+%2B+w%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D+=+%28-b+-+w%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-b+-+w%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-b+%2B+w-b+-+w%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"%28-b+%2B+w-b+-+w%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine the fractions.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28-b-b%29+%2B+%28w-w%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"%28%28-b-b%29+%2B+%28w-w%29%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Group like terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-2b%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"%28-2b%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms. Notice how the \"w\" terms cancel each other out completely.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-b%2Fa ALIGN=MIDDLE  ALT=\"-b%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce\r<BR>\n" );
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document.write( "So this shows us that the sum of the roots of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-b%2Fa ALIGN=MIDDLE  ALT=\"-b%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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