document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=196635>Question 196635</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi!\r<BR>\n" );
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document.write( "On the following problem I keep getting the incorrect answer. I think I'm close to the right answer just making a small error somewhere. Can you help me?\r<BR>\n" );
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document.write( "Solve the equation for the variable x. The constant a represents a positive real number. \r<BR>\n" );
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document.write( "a^3x^3 + 64 = 0  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #147376 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=147376\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E3x%5E3+%2B+64+=+0 ALIGN=MIDDLE  ALT=\"a%5E3x%5E3+%2B+64+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28ax%29%5E3+%2B+64+=+0 ALIGN=MIDDLE  ALT=\"%28ax%29%5E3+%2B+64+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E3x%5E3 ALIGN=MIDDLE  ALT=\"a%5E3x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28ax%29%5E3 ALIGN=MIDDLE  ALT=\"%28ax%29%5E3\"   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=%28ax%29%5E3+%2B+4%5E3+=+0 ALIGN=MIDDLE  ALT=\"%28ax%29%5E3+%2B+4%5E3+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=64 ALIGN=MIDDLE  ALT=\"64\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284%29%5E3 ALIGN=MIDDLE  ALT=\"%284%29%5E3\"   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=%28ax%2B4%29%28%28ax%29%5E2-%28ax%29%284%29%2B%284%29%5E2%29+=+0 ALIGN=MIDDLE  ALT=\"%28ax%2B4%29%28%28ax%29%5E2-%28ax%29%284%29%2B%284%29%5E2%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor the left side using the sum of cubes formula\r<BR>\n" );
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document.write( "Recall, the sum of cubes formula is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5E3%2BB%5E3=%28A%2BB%29%28A%5E2-AB%2BB%5E2%29 ALIGN=MIDDLE  ALT=\"A%5E3%2BB%5E3=%28A%2BB%29%28A%5E2-AB%2BB%5E2%29\"   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=%28ax%2B4%29%28a%5E2x%5E2-4ax%2B16%29+=+0 ALIGN=MIDDLE  ALT=\"%28ax%2B4%29%28a%5E2x%5E2-4ax%2B16%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply and simplify\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%2B4=0 ALIGN=MIDDLE  ALT=\"ax%2B4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2x%5E2-4ax%2B16=0 ALIGN=MIDDLE  ALT=\"a%5E2x%5E2-4ax%2B16=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Use the zero product property to break up the factors\r<BR>\n" );
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document.write( "-----------------------------------------------------\r<BR>\n" );
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document.write( "Let's solve the first equation: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%2B4=0 ALIGN=MIDDLE  ALT=\"ax%2B4=0\"   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=ax%2B4=0 ALIGN=MIDDLE  ALT=\"ax%2B4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the first equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax=-4 ALIGN=MIDDLE  ALT=\"ax=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 4 from both sides.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-4%2Fa ALIGN=MIDDLE  ALT=\"x=-4%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by \"a\" to isolate \"x\".\r<BR>\n" );
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document.write( "So the first solution is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-4%2Fa ALIGN=MIDDLE  ALT=\"x=-4%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "-----------------------------------------------------\r<BR>\n" );
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document.write( "Now let's solve the second equation: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2x%5E2-4ax%2B16=0 ALIGN=MIDDLE  ALT=\"a%5E2x%5E2-4ax%2B16=0\"   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=a%5E2x%5E2-4ax%2B16=0 ALIGN=MIDDLE  ALT=\"a%5E2x%5E2-4ax%2B16=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the second equation.\r<BR>\n" );
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document.write( "Notice we have a quadratic in the form of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Ax%5E2%2BBx%2BC ALIGN=MIDDLE  ALT=\"Ax%5E2%2BBx%2BC\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=a%5E2 ALIGN=MIDDLE  ALT=\"A=a%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B=-4a ALIGN=MIDDLE  ALT=\"B=-4a\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=16 ALIGN=MIDDLE  ALT=\"C=16\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's use the quadratic formula to solve for \"x\"\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-B+%2B-+sqrt%28+B%5E2-4AC+%29%29%2F%282A%29 ALIGN=MIDDLE  ALT=\"x+=+%28-B+%2B-+sqrt%28+B%5E2-4AC+%29%29%2F%282A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the quadratic formula\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-%28-4a%29+%2B-+sqrt%28+%28-4a%29%5E2-4%28a%5E2%29%2816%29+%29%29%2F%282%28a%5E2%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=a%5E2 ALIGN=MIDDLE  ALT=\"A=a%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B=-4a ALIGN=MIDDLE  ALT=\"B=-4a\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=16 ALIGN=MIDDLE  ALT=\"C=16\"   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=x+=+%284a+%2B-+sqrt%28+%28-4a%29%5E2-4%28a%5E2%29%2816%29+%29%29%2F%282%28a%5E2%29%29 ALIGN=MIDDLE  ALT=\"x+=+%284a+%2B-+sqrt%28+%28-4a%29%5E2-4%28a%5E2%29%2816%29+%29%29%2F%282%28a%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Negate -4a to get 4a\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%284a+%2B-+sqrt%28+16a%5E2-4%28a%5E2%29%2816%29+%29%29%2F%282%28a%5E2%29%29 ALIGN=MIDDLE  ALT=\"x+=+%284a+%2B-+sqrt%28+16a%5E2-4%28a%5E2%29%2816%29+%29%29%2F%282%28a%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square -4a to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16a%5E2 ALIGN=MIDDLE  ALT=\"16a%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=x+=+%284a+%2B-+sqrt%28+16a%5E2-64a%5E2+%29%29%2F%282a%5E2%29 ALIGN=MIDDLE  ALT=\"x+=+%284a+%2B-+sqrt%28+16a%5E2-64a%5E2+%29%29%2F%282a%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%284a+%2B-+sqrt%28+-48a%5E2+%29%29%2F%282a%5E2%29 ALIGN=MIDDLE  ALT=\"x+=+%284a+%2B-+sqrt%28+-48a%5E2+%29%29%2F%282a%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%284a+%2B-+sqrt%28+-1%2A16%2A3%2Aa%5E2+%29%29%2F%282a%5E2%29 ALIGN=MIDDLE  ALT=\"x+=+%284a+%2B-+sqrt%28+-1%2A16%2A3%2Aa%5E2+%29%29%2F%282a%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor -48 into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2A16%2A3 ALIGN=MIDDLE  ALT=\"-1%2A16%2A3\"   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=x+=+%284a+%2B-+sqrt%28-1%29%2Asqrt%2816%29%2Asqrt%283%29%2Asqrt%28a%5E2%29+%29%2F%282a%5E2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the square root.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%284a+%2B-+i%2A4%2Asqrt%283%29%2Aa+%29%2F%282a%5E2%29 ALIGN=MIDDLE  ALT=\"x+=+%284a+%2B-+i%2A4%2Asqrt%283%29%2Aa+%29%2F%282a%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify the square roots and replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28-1%29 ALIGN=MIDDLE  ALT=\"sqrt%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > with \"i\".\r<BR>\n" );
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document.write( "Note: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i=sqrt%28-1%29 ALIGN=MIDDLE  ALT=\"i=sqrt%28-1%29\"   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=x+=+%284a+%2B-+4a%2Asqrt%283%29%2Ai+%29%2F%282a%5E2%29 ALIGN=MIDDLE  ALT=\"x+=+%284a+%2B-+4a%2Asqrt%283%29%2Ai+%29%2F%282a%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange the terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%284a+%2B+4a%2Asqrt%283%29%2Ai+%29%2F%282a%5E2%29 ALIGN=MIDDLE  ALT=\"x+=+%284a+%2B+4a%2Asqrt%283%29%2Ai+%29%2F%282a%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+%284a+-+4a%2Asqrt%283%29%2Ai+%29%2F%282a%5E2%29 ALIGN=MIDDLE  ALT=\"a+=+%284a+-+4a%2Asqrt%283%29%2Ai+%29%2F%282a%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the \"plus/minus\"\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%282+%2B+2%2Asqrt%283%29%2Ai+%29%2Fa ALIGN=MIDDLE  ALT=\"x+=+%282+%2B+2%2Asqrt%283%29%2Ai+%29%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%282+-+2%2Asqrt%283%29%2Ai+%29%2Fa ALIGN=MIDDLE  ALT=\"x+=+%282+-+2%2Asqrt%283%29%2Ai+%29%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce\r<BR>\n" );
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document.write( "So the next two solutions are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%282+%2B+2%2Asqrt%283%29%2Ai+%29%2Fa ALIGN=MIDDLE  ALT=\"x+=+%282+%2B+2%2Asqrt%283%29%2Ai+%29%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%282+-+2%2Asqrt%283%29%2Ai+%29%2Fa ALIGN=MIDDLE  ALT=\"x+=+%282+-+2%2Asqrt%283%29%2Ai+%29%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "================================================\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the three solutions are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-4%2Fa ALIGN=MIDDLE  ALT=\"x=-4%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%282+%2B+2%2Asqrt%283%29%2Ai+%29%2Fa ALIGN=MIDDLE  ALT=\"x+=+%282+%2B+2%2Asqrt%283%29%2Ai+%29%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%282+-+2%2Asqrt%283%29%2Ai+%29%2Fa ALIGN=MIDDLE  ALT=\"x+=+%282+-+2%2Asqrt%283%29%2Ai+%29%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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