document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=59156>Question 59156</A>: <I><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+105a%5E3b%2B27a%5E2b%5E2-33ab%5E3+ ALIGN=MIDDLE  ALT=\"+105a%5E3b%2B27a%5E2b%5E2-33ab%5E3+\"   DISABLED_style= \"max-width:90%; height:auto;\" >  Can anyone help me solve this and more importantly, show me the process and some of the rules. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #40556 by </B><A HREF=/tutors/aboutme.mpl?userid=fanks><B>fanks(6)</B></A><img src=\"/images/stars/stars-05.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=fanks><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=40556\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Does it equals zero? In that case...\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+105a%5E3b%2B27a%5E2b%5E2-33ab%5E3+=+3ab+%2835a%5E2%2B9ab-11b%5E2%29+=+0+ ALIGN=MIDDLE  ALT=\"+105a%5E3b%2B27a%5E2b%5E2-33ab%5E3+=+3ab+%2835a%5E2%2B9ab-11b%5E2%29+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "From here (from the first multiplier) we can see that one solution is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+0 ALIGN=MIDDLE  ALT=\"a+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, another is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+0 ALIGN=MIDDLE  ALT=\"b+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "What solutions are there from the second multiplier (from the second brackets)?<BR>\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+b+%3C%3E+0+ ALIGN=MIDDLE  ALT=\"+b+%3C%3E+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+b%5E2+ ALIGN=MIDDLE  ALT=\"+b%5E2+\"   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=+35%28a%2Fb%29%5E2+%2B+9%28a%2Fb%29+-+11+=+0+ ALIGN=MIDDLE  ALT=\"+35%28a%2Fb%29%5E2+%2B+9%28a%2Fb%29+-+11+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Here we get a quadratic equation, let the quadratic equation solver solve this for us:\r<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=solve_quadratic_equation.solver>SOLVE quadratic equation (work shown, graph etc)</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=35x%5E2%2B9x%2B-11+=+0 ALIGN=MIDDLE  ALT=\"35x%5E2%2B9x%2B-11+=+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" );
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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" );
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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=%289%29%5E2-4%2A35%2A-11=1621 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%289%29%5E2-4%2A35%2A-11=1621\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "  Discriminant d=1621 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+=+%28-9%2B-sqrt%28+1621+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28-9%2B-sqrt%28+1621+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D+=+%28-%289%29%2Bsqrt%28+1621+%29%29%2F2%5C35+=+0.446594918262783 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%289%29%2Bsqrt%28+1621+%29%29%2F2%5C35+=+0.446594918262783\"   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-%289%29-sqrt%28+1621+%29%29%2F2%5C35+=+-0.70373777540564 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%289%29-sqrt%28+1621+%29%29%2F2%5C35+=+-0.70373777540564\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "    Quadratic expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=35x%5E2%2B9x%2B-11 ALIGN=MIDDLE  ALT=\"35x%5E2%2B9x%2B-11\"   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=35x%5E2%2B9x%2B-11+=+%28x-0.446594918262783%29%2A%28x--0.70373777540564%29 ALIGN=MIDDLE  ALT=\"35x%5E2%2B9x%2B-11+=+%28x-0.446594918262783%29%2A%28x--0.70373777540564%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 0.446594918262783, -0.70373777540564.\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+35%2Ax%5E2%2B9%2Ax%2B-11+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+35%2Ax%5E2%2B9%2Ax%2B-11+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( "So, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a%2Fb+=+%28-9+%2B+sqrt+%281621%29%29+%2F+70 ALIGN=MIDDLE  ALT=\"+a%2Fb+=+%28-9+%2B+sqrt+%281621%29%29+%2F+70\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a%2Fb+=+%28-9+-+sqrt+%281621%29%29+%2F+70 ALIGN=MIDDLE  ALT=\"+a%2Fb+=+%28-9+-+sqrt+%281621%29%29+%2F+70\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a+=+b%28-9+%2B+sqrt+%281621%29%29+%2F+70 ALIGN=MIDDLE  ALT=\"+a+=+b%28-9+%2B+sqrt+%281621%29%29+%2F+70\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a+=+b%28-9+-+sqrt+%281621%29%29+%2F+70 ALIGN=MIDDLE  ALT=\"+a+=+b%28-9+-+sqrt+%281621%29%29+%2F+70\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Meaning that you can take very (infinitly) many different \"a\"s and \"b\"s as long as they have one of these two proportions.\r<BR>\n" );
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document.write( "So, along with the two zero cases (at the beginning), this is the solution of the given equation. </SPAN>\n" );
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