document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=691290>Question 691290</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use a calculator to find all solutions in the interval (0, 2&#960;). Round the answers to two decimal places. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.)<BR>\n" );
document.write( "5 cos3 x &#8722; 20 cos2 x + cos x &#8722; 4 = 0<BR>\n" );
document.write( "  Hint: Factor by grouping. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #426661 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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=426661\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> (In the future please use \"^\" to indicate exponents. Otherwise it is impossible to tell the difference between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%5E3%28x%29 ALIGN=MIDDLE  ALT=\"cos%5E3%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and cos(3x). I was not going to even try this problem because I could not tell the difference. It is only the \"factor by grouping\" that helped me figure out that the 3 and the 2 must be exponents.)<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5cos%5E3%28x%29+-+20cos%5E2%28x%29+%2B+cos%28x%29+-+4+=+0 ALIGN=MIDDLE  ALT=\"5cos%5E3%28x%29+-+20cos%5E2%28x%29+%2B+cos%28x%29+-+4+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Factoring by grouping...<BR>\n" );
document.write( "Group the first two terms and the last two terms:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285cos%5E3%28x%29+-+20cos%5E2%28x%29%29+%2B+%28cos%28x%29+-+4%29+=+0 ALIGN=MIDDLE  ALT=\"%285cos%5E3%28x%29+-+20cos%5E2%28x%29%29+%2B+%28cos%28x%29+-+4%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Factor out the Greatest Common Factor (GCF) of each group. (Note: This is one of the rare cases when one actually factors out a GCF of 1!)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5cos%5E2%28x%29%28cos%28x%29+-+4%29+%2B+1%28cos%28x%29+-+4%29+=+0 ALIGN=MIDDLE  ALT=\"5cos%5E2%28x%29%28cos%28x%29+-+4%29+%2B+1%28cos%28x%29+-+4%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "As we can see we have a common factor between the two groups: cos(x) - 4. We can factor out this common factor:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285cos%5E2%28x%29+%2B+1%29%28cos%28x%29+-+4%29+=+0 ALIGN=MIDDLE  ALT=\"%285cos%5E2%28x%29+%2B+1%29%28cos%28x%29+-+4%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "From the Zero Product Property:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5cos%5E2%28x%29+%2B+1+=+0 ALIGN=MIDDLE  ALT=\"5cos%5E2%28x%29+%2B+1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29+-+4+=+0 ALIGN=MIDDLE  ALT=\"cos%28x%29+-+4+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Solving the first one...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5cos%5E2%28x%29+=+-1 ALIGN=MIDDLE  ALT=\"5cos%5E2%28x%29+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%5E2%28x%29+=+-1%2F5 ALIGN=MIDDLE  ALT=\"cos%5E2%28x%29+=+-1%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "At this point we should realize that there is no solution to this equation. Squaring a cos (or anything for that matter) cannot result in a negative number.<br><BR>\n" );
document.write( "Solving the second equation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29+-+4+=+0 ALIGN=MIDDLE  ALT=\"cos%28x%29+-+4+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29+=+4 ALIGN=MIDDLE  ALT=\"cos%28x%29+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Again we should recognize that there is no solution. A cos cannot be larger than 1!<br><BR>\n" );
document.write( "So your equation has no solutions! </SPAN>\n" );
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