document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1028991>Question 1028991</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find all the values of x in the interval [0,2pi] such that the gradient of the curve y=12sinx-5cosx is 13/2. (using differentiation) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #644079 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=644079\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Find all the values of x in the interval [0,2pi] such that the gradient of the curve y=12sinx-5cosx is 13/2.<BR>\n" );
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document.write( "y' = 12cos(x) + 5sin(x) = 13/2<BR>\n" );
document.write( "12*sqrt(1 - sin^2) = -5sin + 13/2<BR>\n" );
document.write( "144 - 144sin^2 = 25sin^2 - 65sin + 169/4<BR>\n" );
document.write( "169sin^2 - 65sin - 407/4 = 0<BR>\n" );
document.write( "-------<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=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=169x%5E2%2B-65x%2B-101.75+=+0 ALIGN=MIDDLE  ALT=\"169x%5E2%2B-65x%2B-101.75+=+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=%28-65%29%5E2-4%2A169%2A-101.75=73008 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-65%29%5E2-4%2A169%2A-101.75=73008\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=73008 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--65%2B-sqrt%28+73008+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--65%2B-sqrt%28+73008+%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-%28-65%29%2Bsqrt%28+73008+%29%29%2F2%5C169+=+0.991715757339482 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28-65%29%2Bsqrt%28+73008+%29%29%2F2%5C169+=+0.991715757339482\"   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-%28-65%29-sqrt%28+73008+%29%29%2F2%5C169+=+-0.607100372724097 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28-65%29-sqrt%28+73008+%29%29%2F2%5C169+=+-0.607100372724097\"   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=169x%5E2%2B-65x%2B-101.75 ALIGN=MIDDLE  ALT=\"169x%5E2%2B-65x%2B-101.75\"   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=169x%5E2%2B-65x%2B-101.75+=+%28x-0.991715757339482%29%2A%28x--0.607100372724097%29 ALIGN=MIDDLE  ALT=\"169x%5E2%2B-65x%2B-101.75+=+%28x-0.991715757339482%29%2A%28x--0.607100372724097%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 0.991715757339482, -0.607100372724097.\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+169%2Ax%5E2%2B-65%2Ax%2B-101.75+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+169%2Ax%5E2%2B-65%2Ax%2B-101.75+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( "==================<BR>\n" );
document.write( "sin(x) = 0.991715757339482<BR>\n" );
document.write( "x =~ 82.62 degs, 97.38 degs<BR>\n" );
document.write( "===========<BR>\n" );
document.write( "sin(x) = -0.607100372724097<BR>\n" );
document.write( "x =~ 217.38 degs, 322.62 degs<BR>\n" );
document.write( "=================<BR>\n" );
document.write( "Convert those to radians.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );