document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=172642>Question 172642</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi!\r<BR>\n" );
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document.write( "I have (cos(4X))^2 + (sqrt(3)/2) * sin(4X) = 0<BR>\n" );
document.write( "Anyone out there that can help me? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #130653 by </B><A HREF=/tutors/aboutme.mpl?userid=eggsarecool><B>eggsarecool(46)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=eggsarecool><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=130653\">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=%28cos%284X%29%29%5E2+%2B+%28sqrt%283%29%2F2%29+%2A+sin%284X%29+=+0 ALIGN=MIDDLE  ALT=\"%28cos%284X%29%29%5E2+%2B+%28sqrt%283%29%2F2%29+%2A+sin%284X%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Ok so the first thing to remember is that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28cos%28x%29%29%5E2=1-%28sin%28x%29%29%5E2 ALIGN=MIDDLE  ALT=\"%28cos%28x%29%29%5E2=1-%28sin%28x%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28cos%284X%29%29%5E2=1-%28sin%284X%29%29%5E2 ALIGN=MIDDLE  ALT=\"%28cos%284X%29%29%5E2=1-%28sin%284X%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So we now rewrite your equation as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1-%28sin%284X%29%29%5E2+%2B+%28sqrt%283%29%2F2%29+%2A+sin%284X%29+=+0 ALIGN=MIDDLE  ALT=\"1-%28sin%284X%29%29%5E2+%2B+%28sqrt%283%29%2F2%29+%2A+sin%284X%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which is the same as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%28sin%284X%29%29%5E2+%2B+%28sqrt%283%29%2F2%29+%2A+sin%284X%29+%2B+1+=+0 ALIGN=MIDDLE  ALT=\"-%28sin%284X%29%29%5E2+%2B+%28sqrt%283%29%2F2%29+%2A+sin%284X%29+%2B+1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now set <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=sin%284x%29 ALIGN=MIDDLE  ALT=\"y=sin%284x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > this also means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2=%28sin%284X%29%29%5E2 ALIGN=MIDDLE  ALT=\"y%5E2=%28sin%284X%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Your new equation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-y%5E2+%2B+%28sqrt%283%29%2F2%29y+%2B+1+=+0 ALIGN=MIDDLE  ALT=\"-y%5E2+%2B+%28sqrt%283%29%2F2%29y+%2B+1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This can now be solved using the quadratic formula a=-1, b=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%283%29%2F2%29 ALIGN=MIDDLE  ALT=\"%28sqrt%283%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, c=1\r<BR>\n" );
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document.write( "The formula is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"y+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So you have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-%28sqrt%283%29%2F2%29+%2B-+sqrt%28+%28%28sqrt%283%29%2F2%29%29%5E2-4%2A%28-1%29%2A1+%29%29%2F%282%2A%28-1%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\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=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=ay%5E2%2Bby%2Bc=0 ALIGN=MIDDLE  ALT=\"ay%5E2%2Bby%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1y%5E2%2B0.866025403784439y%2B1+=+0 ALIGN=MIDDLE  ALT=\"-1y%5E2%2B0.866025403784439y%2B1+=+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=y%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"y%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=%280.866025403784439%29%5E2-4%2A-1%2A1=4.75 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%280.866025403784439%29%5E2-4%2A-1%2A1=4.75\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=4.75 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+=+%28-0.866025403784439%2B-sqrt%28+4.75+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28-0.866025403784439%2B-sqrt%28+4.75+%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=y%5B1%5D+=+%28-%280.866025403784439%29%2Bsqrt%28+4.75+%29%29%2F2%5C-1+=+-0.656712033992949 ALIGN=MIDDLE  ALT=\"y%5B1%5D+=+%28-%280.866025403784439%29%2Bsqrt%28+4.75+%29%29%2F2%5C-1+=+-0.656712033992949\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B2%5D+=+%28-%280.866025403784439%29-sqrt%28+4.75+%29%29%2F2%5C-1+=+1.52273743777739 ALIGN=MIDDLE  ALT=\"y%5B2%5D+=+%28-%280.866025403784439%29-sqrt%28+4.75+%29%29%2F2%5C-1+=+1.52273743777739\"   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=-1y%5E2%2B0.866025403784439y%2B1 ALIGN=MIDDLE  ALT=\"-1y%5E2%2B0.866025403784439y%2B1\"   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=-1y%5E2%2B0.866025403784439y%2B1+=+-1%28y--0.656712033992949%29%2A%28y-1.52273743777739%29 ALIGN=MIDDLE  ALT=\"-1y%5E2%2B0.866025403784439y%2B1+=+-1%28y--0.656712033992949%29%2A%28y-1.52273743777739%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: -0.656712033992949, 1.52273743777739.\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%2B0.866025403784439%2Ax%2B1+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-1%2Ax%5E2%2B0.866025403784439%2Ax%2B1+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( "The solver will give you the solutions to the equation<BR>\n" );
document.write( "y1=-0.656712033992949<BR>\n" );
document.write( "y2=1.52273743777739<BR>\n" );
document.write( "But there is one more step, to make the problem clearer we set <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=sin%284x%29 ALIGN=MIDDLE  ALT=\"y=sin%284x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so we actually need a solution for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "To do this we replace they <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" >'s we got and solve for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So we have<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%284x%29=-0.656712033992949 ALIGN=MIDDLE  ALT=\"sin%284x%29=-0.656712033992949\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%284x%29=1.52273743777739 ALIGN=MIDDLE  ALT=\"sin%284x%29=1.52273743777739\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Next step<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x=arcsin%28-0.656712033992949%29 ALIGN=MIDDLE  ALT=\"4x=arcsin%28-0.656712033992949%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > use a calculator to compute this<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x=arcsin%281.52273743777739%29 ALIGN=MIDDLE  ALT=\"4x=arcsin%281.52273743777739%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > when you try to evaluate this you will get an error which means that there is no solution to that part. Only the -0.656712033992949 will yield a result\r<BR>\n" );
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document.write( "Then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28arcsin%28-0.656712033992949%29%29%2F4 ALIGN=MIDDLE  ALT=\"x=%28arcsin%28-0.656712033992949%29%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > use a calculator for this two.\r<BR>\n" );
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document.write( "Enjoy :) </SPAN>\n" );
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