document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=145753>Question 145753</A>: <I><SPAN STYLE=\"word-break: break-all;\"> how do I solve: c^3+c^2-7c-3=0, given root -3? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #106366 by </B><A HREF=/tutors/aboutme.mpl?userid=nabla><B>nabla(475)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=nabla><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=106366\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First of all, note that this method is fairly advanced. If you don't understand what I'm doing, E-mail me what method you were supposed to use to solve this...\r<BR>\n" );
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document.write( "I'm going to change c to x.<BR>\n" );
document.write( "By the given information we have,\r<BR>\n" );
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document.write( "(x+3)(ax^2+bx+c)=0\r<BR>\n" );
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document.write( "and if we expand this out, we have:<BR>\n" );
document.write( "3 c + 3 b x + c x + 3 a x^2 + b x^2 + a x^3=0\r<BR>\n" );
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document.write( "From this and the original coefficients 1, 1, -7, -3, respectively, it follows that:<BR>\n" );
document.write( "3c=-3, implies c=-1<BR>\n" );
document.write( "3b+c=-7, implied 3b=-6, implies b=-2<BR>\n" );
document.write( "3a+b=1, implies 3a=3, implies a=1.\r<BR>\n" );
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document.write( "We don't have to solve the cubic coefficients.\r<BR>\n" );
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document.write( "So, the original polynomial factors as:<BR>\n" );
document.write( "(x+3)(x^2-2x-1)=0<BR>\n" );
document.write( "Now, we can solve for the other roots by applying the quadratic formulae:<BR>\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=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2%2B-2x%2B-1+=+0 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-2x%2B-1+=+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=x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"x%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=%28-2%29%5E2-4%2A1%2A-1=8 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-2%29%5E2-4%2A1%2A-1=8\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=8 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--2%2B-sqrt%28+8+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--2%2B-sqrt%28+8+%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=x%5B1%5D+=+%28-%28-2%29%2Bsqrt%28+8+%29%29%2F2%5C1+=+2.41421356237309 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28-2%29%2Bsqrt%28+8+%29%29%2F2%5C1+=+2.41421356237309\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D+=+%28-%28-2%29-sqrt%28+8+%29%29%2F2%5C1+=+-0.414213562373095 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28-2%29-sqrt%28+8+%29%29%2F2%5C1+=+-0.414213562373095\"   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=1x%5E2%2B-2x%2B-1 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-2x%2B-1\"   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=1x%5E2%2B-2x%2B-1+=+1%28x-2.41421356237309%29%2A%28x--0.414213562373095%29 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-2x%2B-1+=+1%28x-2.41421356237309%29%2A%28x--0.414213562373095%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 2.41421356237309, -0.414213562373095.\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+1%2Ax%5E2%2B-2%2Ax%2B-1+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-2%2Ax%2B-1+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( "Which gives <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1%2B-sqrt%282%29 ALIGN=MIDDLE  ALT=\"x=1%2B-sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So, all zeroes are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1%2B-sqrt%282%29 ALIGN=MIDDLE  ALT=\"x=1%2B-sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and -3. We can see this in the following graph of the cubic:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+300%2C+200%2C+-5%2C+5%2C+-5%2C+5%2C+x%5E3%2Bx%5E2-7x-3+%29 ALIGN=MIDDLE  ALT=\"graph%28+300%2C+200%2C+-5%2C+5%2C+-5%2C+5%2C+x%5E3%2Bx%5E2-7x-3+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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