document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=90951>Question 90951</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0132014084/algebrahomeworkh></A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> Solve the equation algebraically.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5Ex%2B2%5E%28-x%29=5 ALIGN=MIDDLE  ALT=\"2%5Ex%2B2%5E%28-x%29=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "I've tried almost everything I can think of, and still no luck! Please help! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #66027 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=66027\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Tricky problem:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5Ex%2B2%5E%28-x%29=5 ALIGN=MIDDLE  ALT=\"2%5Ex%2B2%5E%28-x%29=5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Let's convert the term with the minus exponent to fractional form:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5Ex+%2B+1%2F%282%5Ex%29+=+5 ALIGN=MIDDLE  ALT=\"2%5Ex+%2B+1%2F%282%5Ex%29+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Next, get rid of the denominator by multiplying both sides (all terms) by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5Ex ALIGN=MIDDLE  ALT=\"2%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" >. When<BR>\n" );
document.write( "you do that multiplication the equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%5Ex%2A2%5Ex%29+%2B+2%5Ex%281%2F2%5Ex%29+=+5%2A2%5Ex ALIGN=MIDDLE  ALT=\"%282%5Ex%2A2%5Ex%29+%2B+2%5Ex%281%2F2%5Ex%29+=+5%2A2%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "which becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E%28x%2Bx%29+%2B+cross%282%5Ex%29%2A1%2Fcross%282%5Ex%29+=+5%2A2%5Ex ALIGN=MIDDLE  ALT=\"2%5E%28x%2Bx%29+%2B+cross%282%5Ex%29%2A1%2Fcross%282%5Ex%29+=+5%2A2%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "which simplifies to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E%282x%29%29+%2B+1+=+5%2A2%5Ex ALIGN=MIDDLE  ALT=\"2%5E%282x%29%29+%2B+1+=+5%2A2%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now recognize that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E%282x%29 ALIGN=MIDDLE  ALT=\"2%5E%282x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equal to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%5Ex%29%5E2+ ALIGN=MIDDLE  ALT=\"%282%5Ex%29%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > by the power rule of exponents.<BR>\n" );
document.write( "Substituting this makes the equation become:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%5Ex%29%5E2+%2B+1+=+5%2A2%5Ex ALIGN=MIDDLE  ALT=\"%282%5Ex%29%5E2+%2B+1+=+5%2A2%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Then subtracting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2A%282%5Ex%29+ ALIGN=MIDDLE  ALT=\"5%2A%282%5Ex%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides results in:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%5Ex%29%5E2+-+5%2A%282%5Ex%29+%2B+1+=+0 ALIGN=MIDDLE  ALT=\"%282%5Ex%29%5E2+-+5%2A%282%5Ex%29+%2B+1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5Ex+=+t ALIGN=MIDDLE  ALT=\"2%5Ex+=+t\"   DISABLED_style= \"max-width:90%; height:auto;\" > ... just because it helps to clarify what's coming next. Substituting<BR>\n" );
document.write( "t for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5Ex ALIGN=MIDDLE  ALT=\"2%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > gives:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5E2+-+5t+%2B+1+=+0 ALIGN=MIDDLE  ALT=\"t%5E2+-+5t+%2B+1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Notice that this is just an \"ordinary\" quadratic equation. If you apply the quadratic<BR>\n" );
document.write( "formula to this problem you get two answers for t as follows:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+-%28-5%29%2F%282%2A1%29+%2B-+sqrt%28%28-5%29%5E2+-4%2A1%2A1%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"t+=+-%28-5%29%2F%282%2A1%29+%2B-+sqrt%28%28-5%29%5E2+-4%2A1%2A1%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This simplifies to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+5%2F2+%2B-+sqrt%2825-4%29%2F2+=+5%2F2+%2B-+sqrt%2821%29%2F2 ALIGN=MIDDLE  ALT=\"t+=+5%2F2+%2B-+sqrt%2825-4%29%2F2+=+5%2F2+%2B-+sqrt%2821%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "So we have two possible answers for t:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+5%2F2+%2B+sqrt%2821%29%2F2 ALIGN=MIDDLE  ALT=\"t+=+5%2F2+%2B+sqrt%2821%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+5%2F2+-+sqrt%2821%29%2F2 ALIGN=MIDDLE  ALT=\"t+=+5%2F2+-+sqrt%2821%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Let's make life a little easier by converting these answers to decimals by using a calculator.<BR>\n" );
document.write( "If you do the work you should get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+5%2F2+%2B+sqrt%2821%29%2F2+=+2.5+%2B+4.582575695%2F2=2.5+%2B+2.291287847+=+4.791287847 ALIGN=MIDDLE  ALT=\"t+=+5%2F2+%2B+sqrt%2821%29%2F2+=+2.5+%2B+4.582575695%2F2=2.5+%2B+2.291287847+=+4.791287847\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and <BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+5%2F2+-+sqrt%2821%29%2F2+=+2.5+-+4.582575695%2F2=2.5+-+2.291287847+=+0.208712152 ALIGN=MIDDLE  ALT=\"t+=+5%2F2+-+sqrt%2821%29%2F2+=+2.5+-+4.582575695%2F2=2.5+-+2.291287847+=+0.208712152\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
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document.write( "There they are ... two possible answers for t.  But let's recall that t is equal to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5Ex ALIGN=MIDDLE  ALT=\"2%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "So we can write our two answers as:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5Ex+=+4.791287847 ALIGN=MIDDLE  ALT=\"2%5Ex+=+4.791287847\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5Ex+=+0.208712152 ALIGN=MIDDLE  ALT=\"2%5Ex+=+0.208712152\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now apply logs to solve for x.  We could use any logs of any base, but for grins let's use<BR>\n" );
document.write( "the natural logs ... indicated by ln .... on both equations:<BR>\n" );
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document.write( "Start with the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5Ex+=+4.791287847 ALIGN=MIDDLE  ALT=\"2%5Ex+=+4.791287847\"   DISABLED_style= \"max-width:90%; height:auto;\" > and take the ln of both sides to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%282%5Ex%29+=+ln%284.791287847%29 ALIGN=MIDDLE  ALT=\"ln%282%5Ex%29+=+ln%284.791287847%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "on the left side we can make the exponent x a multiplier to convert the equation to:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Aln%282%29+=+ln%284.791287847%29 ALIGN=MIDDLE  ALT=\"x%2Aln%282%29+=+ln%284.791287847%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now use a calculator to determine that ln(2) = 0.69314718 and ln(4.791287847) = 1.566799237.<BR>\n" );
document.write( "Substituting these values for the logarithms results in the equation becoming:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2A%280.69314718%29+=+1.566799237 ALIGN=MIDDLE  ALT=\"x%2A%280.69314718%29+=+1.566799237\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Divide both sides by 0.6931471 and you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+1.566799237%2F0.6931471+=+2.260413752 ALIGN=MIDDLE  ALT=\"x+=+1.566799237%2F0.6931471+=+2.260413752\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "One possible value for x<BR>\n" );
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document.write( "The other value for x is basically worked the same way ... just different numbers. Start<BR>\n" );
document.write( "with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5Ex+=+0.208712152 ALIGN=MIDDLE  ALT=\"2%5Ex+=+0.208712152\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "Take natural logs of both sides:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%282%5Ex%29+=+ln%280.208712152%29 ALIGN=MIDDLE  ALT=\"ln%282%5Ex%29+=+ln%280.208712152%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Bring the exponent out:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Aln%282%29=+ln%280.208712152%29 ALIGN=MIDDLE  ALT=\"x%2Aln%282%29=+ln%280.208712152%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Use a calculator to find the two natural logs:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2A%280.69314718%29=+-1.566799239 ALIGN=MIDDLE  ALT=\"x%2A%280.69314718%29=+-1.566799239\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Divide both sides by 0.69314718<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-1.566799239%2F0.69314718+=+-2.260413495 ALIGN=MIDDLE  ALT=\"x+=+-1.566799239%2F0.69314718+=+-2.260413495\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Check every step of the math above just to make sure I didn't read the calculator <BR>\n" );
document.write( "wrong somewhere.  I only know that x = 2.260413 and x = -2.260413 are pretty close to correct.<BR>\n" );
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document.write( "Hope this helps you plow through the problem OK. Time for a coffee break.<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );