document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1057009>Question 1057009</A>: <I><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5E3-13x%2B12=0+ ALIGN=MIDDLE  ALT=\"+x%5E3-13x%2B12=0+\"   DISABLED_style= \"max-width:90%; height:auto;\" > find the answers of x without using a calculator and please if you have a way to solve the cubic equations put it i still have 30 more to go </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #672089 by </B><A HREF=/tutors/aboutme.mpl?userid=solve_for_x><B>solve_for_x(190)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solve_for_x><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=672089\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Unfortunately, the direct solution formula for a cubic equation is very, very ugly.\r<BR>\n" );
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document.write( "The better way to solve something like this is to find one solution that works, and <BR>\n" );
document.write( "then use that to factor a term out of the expression, which will reduce the rest of<BR>\n" );
document.write( "it to a quadratic expression. You can then either factor the quadratic, or use the <BR>\n" );
document.write( "quadratic formula.\r<BR>\n" );
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document.write( "For this problem, it is pretty easy to see that x = 1 will be a solution, since it gives:\r<BR>\n" );
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document.write( "1 - 13(1) + 12 = 0\r<BR>\n" );
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document.write( "1 - 13 + 12 = 0\r<BR>\n" );
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document.write( "-12 + 12 = 0\r<BR>\n" );
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document.write( "0 = 0\r<BR>\n" );
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document.write( "Since x = 1 is a solution, you know that (x - 1) is a factor.\r<BR>\n" );
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document.write( "Using polynomial division, divide <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3+-+13x+%2B+12 ALIGN=MIDDLE  ALT=\"x%5E3+-+13x+%2B+12\"   DISABLED_style= \"max-width:90%; height:auto;\" > by (x - 1), to get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+x+-+12 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+x+-+12\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Thus, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3+-+13x+%2B+12+=+%28x+-+1%29%28x%5E2+%2B+x+-+12%29 ALIGN=MIDDLE  ALT=\"x%5E3+-+13x+%2B+12+=+%28x+-+1%29%28x%5E2+%2B+x+-+12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Next, factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+x+-+12 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+x+-+12\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x+-+3%29%28x+%2B+4%29 ALIGN=MIDDLE  ALT=\"%28x+-+3%29%28x+%2B+4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This means that:\r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3+-+13x+%2B+12+=+%28x+-+1%29%28x+-+3%29%28x+%2B+4%29 ALIGN=MIDDLE  ALT=\"x%5E3+-+13x+%2B+12+=+%28x+-+1%29%28x+-+3%29%28x+%2B+4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "And from the factors on the right hand side, you can see that <BR>\n" );
document.write( "the solutions are <strong>-4, 1, and 3</strong>. </SPAN>\n" );
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