document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=632223>Question 632223</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I'm supposed to find the solutions of x for the equation:\r<BR>\n" );
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document.write( "27x^3=(x+5)^3\r<BR>\n" );
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document.write( "I tried using the difference of cubes where a=(x+5)^3 and b=27x^3 but running through it gave a polynomial to the 6th power, which doesn't do me much good. Please help. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #398138 by </B><A HREF=/tutors/aboutme.mpl?userid=reviewermath><B>reviewermath(1029)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=reviewermath><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=reviewermath><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=398138\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Using the difference of two cubes where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E3+=+%28x%2B5%29%5E3 ALIGN=MIDDLE  ALT=\"a%5E3+=+%28x%2B5%29%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E3=27x%5E3 ALIGN=MIDDLE  ALT=\"b%5E3=27x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E3-b%5E3+=+0+ ALIGN=MIDDLE  ALT=\"a%5E3-b%5E3+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a-b%29%28a%5E2%2Bab%2Bb%5E2%29+=+0 ALIGN=MIDDLE  ALT=\"%28a-b%29%28a%5E2%2Bab%2Bb%5E2%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28x%2B5%29+-+3x%29%2A%28%28x%2B5%29%5E2+%2B+3x%28x%2B5%29+%2B+%283x%29%5E2%29+=+0 ALIGN=MIDDLE  ALT=\"%28%28x%2B5%29+-+3x%29%2A%28%28x%2B5%29%5E2+%2B+3x%28x%2B5%29+%2B+%283x%29%5E2%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-2x+%2B+5%29%2A%28x%5E2+%2B+10x+%2B+25+%2B+3x%5E2+%2B+15x+%2B+9x%5E2%29+=+0 ALIGN=MIDDLE  ALT=\"%28-2x+%2B+5%29%2A%28x%5E2+%2B+10x+%2B+25+%2B+3x%5E2+%2B+15x+%2B+9x%5E2%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-2x+%2B+5%29%2A%2813x%5E2+%2B+25x+%2B+25%29+=+0 ALIGN=MIDDLE  ALT=\"%28-2x+%2B+5%29%2A%2813x%5E2+%2B+25x+%2B+25%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-2x+%2B+5+=+0 ALIGN=MIDDLE  ALT=\"+-2x+%2B+5+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13x%5E2+%2B+25x+%2B+25+=+0 ALIGN=MIDDLE  ALT=\"13x%5E2+%2B+25x+%2B+25+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Solve: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2x+%2B+5+=+0 ALIGN=MIDDLE  ALT=\"-2x+%2B+5+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x+=+5%2F2%29 ALIGN=MIDDLE  ALT=\"highlight%28x+=+5%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Solve: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13x%5E2+%2B+25x+%2B+25+=+0 ALIGN=MIDDLE  ALT=\"13x%5E2+%2B+25x+%2B+25+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<TABLE  BORDER=1>\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=13x%5E2%2B25x%2B25+=+0 ALIGN=MIDDLE  ALT=\"13x%5E2%2B25x%2B25+=+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=%2825%29%5E2-4%2A13%2A25=-675 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%2825%29%5E2-4%2A13%2A25=-675\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  The discriminant -675 is less than zero. That means that there are no solutions among real numbers.<BR><BR>\n" );
document.write( "  If you are a student of advanced school algebra and are aware about <u>imaginary numbers<u>, read on.<BR><BR>\n" );
document.write( "  <BR>\n" );
document.write( "      In the field of imaginary numbers, the square root of -675 is + or - <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28+675%29+=+25.9807621135332 ALIGN=MIDDLE  ALT=\"sqrt%28+675%29+=+25.9807621135332\"   DISABLED_style= \"max-width:90%; height:auto;\" >. <BR>\n" );
document.write( "  <BR>\n" );
document.write( "      The solution is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B12%5D+=+%28-25%2B-+i%2Asqrt%28+-675+%29%29%2F2%5C13+=++%28-25%2B-+i%2A25.9807621135332%29%2F2%5C13+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\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+13%2Ax%5E2%2B25%2Ax%2B25+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+13%2Ax%5E2%2B25%2Ax%2B25+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
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document.write( "There is only one real solution, x = 5/2. <BR>\n" );
document.write( "The other two roots are imaginary numbers as computed above. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );