document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1176692>Question 1176692</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 9^x-2(3^x+2)+81=0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #803905 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=803905\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "We'll apply the rule<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%5Eb%29%5Ec+=+a%5E%28b%2Ac%29 ALIGN=MIDDLE  ALT=\"%28a%5Eb%29%5Ec+=+a%5E%28b%2Ac%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "to say that:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%5Ex+=+%283%5E2%29%5Ex ALIGN=MIDDLE  ALT=\"9%5Ex+=+%283%5E2%29%5Ex\"   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=9%5Ex+=+3%5E%282x%29 ALIGN=MIDDLE  ALT=\"9%5Ex+=+3%5E%282x%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=9%5Ex+=+%283%5Ex%29%5E2 ALIGN=MIDDLE  ALT=\"9%5Ex+=+%283%5Ex%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The equation<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%5Ex-2%283%5Ex%2B2%29%2B81=0 ALIGN=MIDDLE  ALT=\"9%5Ex-2%283%5Ex%2B2%29%2B81=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "then becomes<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%5Ex%29%5E2-2%283%5Ex%2B2%29%2B81=0 ALIGN=MIDDLE  ALT=\"%283%5Ex%29%5E2-2%283%5Ex%2B2%29%2B81=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w+=+3%5Ex ALIGN=MIDDLE  ALT=\"w+=+3%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Replace every copy of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%5Ex ALIGN=MIDDLE  ALT=\"3%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w ALIGN=MIDDLE  ALT=\"w\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "We go from<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%5Ex%29%5E2-2%283%5Ex%2B2%29%2B81=0 ALIGN=MIDDLE  ALT=\"%283%5Ex%29%5E2-2%283%5Ex%2B2%29%2B81=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w%5E2-2%28w%2B2%29%2B81=0 ALIGN=MIDDLE  ALT=\"w%5E2-2%28w%2B2%29%2B81=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "-----------------------------------<BR>\n" );
document.write( "Let's solve for w\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w%5E2-2%28w%2B2%29%2B81=0 ALIGN=MIDDLE  ALT=\"w%5E2-2%28w%2B2%29%2B81=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=w%5E2-2w-4%2B81=0 ALIGN=MIDDLE  ALT=\"w%5E2-2w-4%2B81=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=w%5E2-2w%2B77=0 ALIGN=MIDDLE  ALT=\"w%5E2-2w%2B77=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Apply the quadratic formula<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w+=+%28-b%2Bsqrt%28b%5E2-4ac%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"w+=+%28-b%2Bsqrt%28b%5E2-4ac%29%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w+=+%28-b-sqrt%28b%5E2-4ac%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"w+=+%28-b-sqrt%28b%5E2-4ac%29%29%2F%282a%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=w+=+%28-%28-2%29%2Bsqrt%28%28-2%29%5E2-4%281%29%2877%29%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"w+=+%28-%28-2%29%2Bsqrt%28%28-2%29%5E2-4%281%29%2877%29%29%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w+=+%28-%28-2%29-sqrt%28%28-2%29%5E2-4%281%29%2877%29%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"w+=+%28-%28-2%29-sqrt%28%28-2%29%5E2-4%281%29%2877%29%29%29%2F%282%281%29%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=w+=+%284%2Bsqrt%28-304%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"w+=+%284%2Bsqrt%28-304%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w+=+%284-sqrt%28-304%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"w+=+%284-sqrt%28-304%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We run into a problem. The negative under the square root means we'll have complex numbers. I'll assume your teacher hasn't covered the concept yet (please let me know if otherwise)\r<BR>\n" );
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document.write( "So because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w%5E2-2w%2B77=0 ALIGN=MIDDLE  ALT=\"w%5E2-2w%2B77=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > has complex solutions, then so does the original equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%5Ex-2%283%5Ex%2B2%29%2B81=0 ALIGN=MIDDLE  ALT=\"9%5Ex-2%283%5Ex%2B2%29%2B81=0\"   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 that I think about it further, it's possible that the equation you were given is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%5Ex-2%2A3%5E%28x%2B2%29%2B81=0 ALIGN=MIDDLE  ALT=\"9%5Ex-2%2A3%5E%28x%2B2%29%2B81=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > instead of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%5Ex-2%2A%283%5E%28x%29%2B2%29%2B81=0 ALIGN=MIDDLE  ALT=\"9%5Ex-2%2A%283%5E%28x%29%2B2%29%2B81=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Note how the 2 has moved from being in the exponent and then not in the exponent.\r<BR>\n" );
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document.write( "If the 2 is in the exponent, then,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%5Ex-2%2A3%5E%28x%2B2%29%2B81=0 ALIGN=MIDDLE  ALT=\"9%5Ex-2%2A3%5E%28x%2B2%29%2B81=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%5Ex%29%5E2-2%2A%283%5Ex%29%2A3%5E%282%29%2B81=0 ALIGN=MIDDLE  ALT=\"%283%5Ex%29%5E2-2%2A%283%5Ex%29%2A3%5E%282%29%2B81=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%5Ex%29%5E2-2%2A%283%5Ex%29%2A9%2B81=0 ALIGN=MIDDLE  ALT=\"%283%5Ex%29%5E2-2%2A%283%5Ex%29%2A9%2B81=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=w%5E2-18w%2B81=0 ALIGN=MIDDLE  ALT=\"w%5E2-18w%2B81=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=%28w-9%29%5E2=0 ALIGN=MIDDLE  ALT=\"%28w-9%29%5E2=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=w-9+=+sqrt%280%29 ALIGN=MIDDLE  ALT=\"w-9+=+sqrt%280%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w-9=0 ALIGN=MIDDLE  ALT=\"w-9=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=w=9 ALIGN=MIDDLE  ALT=\"w=9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "If w = 9, then, <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w+=+3%5Ex ALIGN=MIDDLE  ALT=\"w+=+3%5Ex\"   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=9+=+3%5Ex ALIGN=MIDDLE  ALT=\"9+=+3%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%5Ex+=+9 ALIGN=MIDDLE  ALT=\"3%5Ex+=+9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%5Ex+=+3%5E2 ALIGN=MIDDLE  ALT=\"3%5Ex+=+3%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+=+2 ALIGN=MIDDLE  ALT=\"x+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So once again, if the equation you were given was <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%5Ex-2%2A3%5E%28x%2B2%29%2B81=0 ALIGN=MIDDLE  ALT=\"9%5Ex-2%2A3%5E%28x%2B2%29%2B81=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (note the 2 in the exponent), then it leads to the single solution of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+2 ALIGN=MIDDLE  ALT=\"x+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We can check this by plugging x = 2 back into the original equation to get...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%5Ex-2%2A3%5E%28x%2B2%29%2B81=0 ALIGN=MIDDLE  ALT=\"9%5Ex-2%2A3%5E%28x%2B2%29%2B81=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=9%5E2-2%2A3%5E%282%2B2%29%2B81=0 ALIGN=MIDDLE  ALT=\"9%5E2-2%2A3%5E%282%2B2%29%2B81=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=9%5E2-2%2A3%5E4%2B81=0 ALIGN=MIDDLE  ALT=\"9%5E2-2%2A3%5E4%2B81=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=81-2%2A81%2B81=0 ALIGN=MIDDLE  ALT=\"81-2%2A81%2B81=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=81-162%2B81=0 ALIGN=MIDDLE  ALT=\"81-162%2B81=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=-81%2B81=0 ALIGN=MIDDLE  ALT=\"-81%2B81=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=0=0 ALIGN=MIDDLE  ALT=\"0=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We get the same thing on both sides, so the answer is confirmed for that equation.<BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );