document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=980725>Question 980725</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If cos^2 x + cos x = 1. so, sin^12 x + 3 sin^10 x + 3 sin^8 x + sin^6 x = ? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #601855 by </B><A HREF=/tutors/aboutme.mpl?userid=Alan3354><B>Alan3354(69443)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Alan3354><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=Alan3354><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=601855\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If cos^2 x + cos x = 1. so, sin^12 x + 3 sin^10 x + 3 sin^8 x + sin^6 x = ?<BR>\n" );
document.write( " ------------<BR>\n" );
document.write( " Solve for cos(x)<BR>\n" );
document.write( " cos^2 x + cos x = 1<BR>\n" );
document.write( " cos^2 x + cos x - 1 = 0<BR>\n" );
document.write( " -----<BR>\n" );
document.write( " Sub u for cos(x)<BR>\n" );
document.write( " u^2 + u - 1 = 0<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=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=1x%5E2%2B1x%2B-1+=+0 ALIGN=MIDDLE  ALT=\"1x%5E2%2B1x%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=%281%29%5E2-4%2A1%2A-1=5 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%281%29%5E2-4%2A1%2A-1=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=5 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+=+%28-1%2B-sqrt%28+5+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28-1%2B-sqrt%28+5+%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-%281%29%2Bsqrt%28+5+%29%29%2F2%5C1+=+0.618033988749895 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%281%29%2Bsqrt%28+5+%29%29%2F2%5C1+=+0.618033988749895\"   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-%281%29-sqrt%28+5+%29%29%2F2%5C1+=+-1.61803398874989 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%281%29-sqrt%28+5+%29%29%2F2%5C1+=+-1.61803398874989\"   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%2B1x%2B-1 ALIGN=MIDDLE  ALT=\"1x%5E2%2B1x%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%2B1x%2B-1+=+%28x-0.618033988749895%29%2A%28x--1.61803398874989%29 ALIGN=MIDDLE  ALT=\"1x%5E2%2B1x%2B-1+=+%28x-0.618033988749895%29%2A%28x--1.61803398874989%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 0.618033988749895, -1.61803398874989.\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%2B1%2Ax%2B-1+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B1%2Ax%2B-1+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
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document.write( " --------------<BR>\n" );
document.write( " cos(x) = -1/2 + sqrt(5)/2<BR>\n" );
document.write( " ---<BR>\n" );
document.write( " Find sin^2(x)<BR>\n" );
document.write( " sin^2 + cos^2 = 1<BR>\n" );
document.write( " sin^2(x) = 1 - (1/4 - sqrt(5)/2 + 5/4) = -1/2 - sqrt(5)/2<BR>\n" );
document.write( " --------------<BR>\n" );
document.write( " sin^12 x + 3 sin^10 x + 3 sin^8 x + sin^6 x = sin^6(x)*(sin^6 + 3sin^4 + 3sin^2 + 1)<BR>\n" );
document.write( " = sin^6(x)*(sin^2 + 1)^3<BR>\n" );
document.write( "------------<BR>\n" );
document.write( "sin^6 = (sin^2)^3 = (1 + sqrt(5))/2<BR>\n" );
document.write( "==============<BR>\n" );
document.write( "(1/2)*(1 + sqrt(5))*(-1/2)*(1 - sqrt(5))<BR>\n" );
document.write( "= -(1/4)*(1 - 5)<BR>\n" );
document.write( "= 1<BR>\n" );
document.write( "======================<BR>\n" );
document.write( "Do the same for the other root, cos(x) = -1/2 - sqrt(5)/2 </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );