document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=155348>Question 155348</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0132402866/algebrahomeworkh>College Algebra</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> Please help me solve this.  I need to put it in standard form then solve for x using the quadratic equation which should give me a complex number as an answer.  The problem is:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-8=0 ALIGN=MIDDLE  ALT=\"x%5E3-8=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Thanks. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #114397 by </B><A HREF=/tutors/aboutme.mpl?userid=nerdybill><B>nerdybill(7384)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=nerdybill><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=nerdybill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=114397\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Notice that your equation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-8=0 ALIGN=MIDDLE  ALT=\"x%5E3-8=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "is a \"difference of two cubes\"<BR>\n" );
document.write( "That is, both terms have cube roots:<BR>\n" );
document.write( "the cube root of x^3 is x<BR>\n" );
document.write( "the cube root of 8^3 is 2<BR>\n" );
document.write( ".<BR>\n" );
document.write( "This site describes these \"special cases\":<BR>\n" );
document.write( "http://www.purplemath.com/modules/specfact2.htm<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Bottom-line:<BR>\n" );
document.write( "If you see a situation of a \"difference of two cubes\", you can factor thus:<BR>\n" );
document.write( "a^3 – b^3 = (a – b)(a^2 + ab + b^2)<BR>\n" );
document.write( ".<BR>\n" );
document.write( "In our case:<BR>\n" );
document.write( "a is x<BR>\n" );
document.write( "b is 2<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Therefore, we can rewrite:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-8=0 ALIGN=MIDDLE  ALT=\"x%5E3-8=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-2%29%28x%5E2+%2B+2x+%2B+2%5E2%29=0 ALIGN=MIDDLE  ALT=\"%28x-2%29%28x%5E2+%2B+2x+%2B+2%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=%28x-2%29%28x%5E2+%2B+2x+%2B+4%29=0 ALIGN=MIDDLE  ALT=\"%28x-2%29%28x%5E2+%2B+2x+%2B+4%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Finally, to solve, we set each factor on the left to zero:<BR>\n" );
document.write( "First term:<BR>\n" );
document.write( "(x-2) = 0<BR>\n" );
document.write( "x = 2 (Here's one \"real\" solution)<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Second term:<BR>\n" );
document.write( "(x^2 + 2x + 4)=0<BR>\n" );
document.write( "Here's where we need to apply the \"quadratic equation\":<BR>\n" );
document.write( "Note: you will find that there are no real solutions, only 2 imaginary ones:\r<BR>\n" );
document.write( "\n" );
document.write( "<TABLE  BORDER=1>\n" );
document.write( "\n" );
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%2B2x%2B4+=+0 ALIGN=MIDDLE  ALT=\"1x%5E2%2B2x%2B4+=+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=%282%29%5E2-4%2A1%2A4=-12 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%282%29%5E2-4%2A1%2A4=-12\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  The discriminant -12 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 -12 is + or - <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28+12%29+=+3.46410161513775 ALIGN=MIDDLE  ALT=\"sqrt%28+12%29+=+3.46410161513775\"   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+=+%28-2%2B-+i%2Asqrt%28+-12+%29%29%2F2%5C1+=++%28-2%2B-+i%2A3.46410161513775%29%2F2%5C1+ 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+1%2Ax%5E2%2B2%2Ax%2B4+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B2%2Ax%2B4+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );