document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=202564>Question 202564</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 2 times the square root of x minus 1 minus the square root of 3x minus 5 equals the square root of x minus 9 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #152760 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=152760\">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=2%2Asqrt%28x-1%29-sqrt%283x-5%29=sqrt%28x-9%29 ALIGN=MIDDLE  ALT=\"2%2Asqrt%28x-1%29-sqrt%283x-5%29=sqrt%28x-9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2Asqrt%28x-1%29-sqrt%283x-5%29%29%5E2=%28sqrt%28x-9%29%29%5E2 ALIGN=MIDDLE  ALT=\"%282%2Asqrt%28x-1%29-sqrt%283x-5%29%29%5E2=%28sqrt%28x-9%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2Asqrt%28x-1%29-sqrt%283x-5%29%29%5E2=x-9 ALIGN=MIDDLE  ALT=\"%282%2Asqrt%28x-1%29-sqrt%283x-5%29%29%5E2=x-9\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square the square root.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28x-1%29-4%2Asqrt%28x-1%29%2Asqrt%283x-5%29%2B3x-5=x-9 ALIGN=MIDDLE  ALT=\"4%28x-1%29-4%2Asqrt%28x-1%29%2Asqrt%283x-5%29%2B3x-5=x-9\"   DISABLED_style= \"max-width:90%; height:auto;\" > FOIL\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x-4-4%2Asqrt%28x-1%29%2Asqrt%283x-5%29%2B3x-5=x-9 ALIGN=MIDDLE  ALT=\"4x-4-4%2Asqrt%28x-1%29%2Asqrt%283x-5%29%2B3x-5=x-9\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7x-9-4%2Asqrt%28x-1%29%2Asqrt%283x-5%29=x-9 ALIGN=MIDDLE  ALT=\"7x-9-4%2Asqrt%28x-1%29%2Asqrt%283x-5%29=x-9\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2Asqrt%28x-1%29%2Asqrt%283x-5%29=x-9%2B9-7x ALIGN=MIDDLE  ALT=\"-4%2Asqrt%28x-1%29%2Asqrt%283x-5%29=x-9%2B9-7x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Get all the non-radical terms to one side\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2Asqrt%28x-1%29%2Asqrt%283x-5%29=-6x ALIGN=MIDDLE  ALT=\"-4%2Asqrt%28x-1%29%2Asqrt%283x-5%29=-6x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2Asqrt%28%28x-1%29%283x-5%29%29=-6x ALIGN=MIDDLE  ALT=\"-4%2Asqrt%28%28x-1%29%283x-5%29%29=-6x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine the roots.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%28x-1%29%283x-5%29%29=%28-6x%29%2F%28-4%29 ALIGN=MIDDLE  ALT=\"sqrt%28%28x-1%29%283x-5%29%29=%28-6x%29%2F%28-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by -4.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%28x-1%29%283x-5%29%29=%283%2F2%29x ALIGN=MIDDLE  ALT=\"sqrt%28%28x-1%29%283x-5%29%29=%283%2F2%29x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%28%28x-1%29%283x-5%29%29%29%5E2=%28%283%2F2%29x%29%5E2 ALIGN=MIDDLE  ALT=\"%28sqrt%28%28x-1%29%283x-5%29%29%29%5E2=%28%283%2F2%29x%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%283x-5%29=%28%283%2F2%29x%29%5E2 ALIGN=MIDDLE  ALT=\"%28x-1%29%283x-5%29=%28%283%2F2%29x%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square the square root\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%283x-5%29=%289%2F4%29x%5E2 ALIGN=MIDDLE  ALT=\"%28x-1%29%283x-5%29=%289%2F4%29x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square the right side\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28x-1%29%283x-5%29=9x%5E2 ALIGN=MIDDLE  ALT=\"4%28x-1%29%283x-5%29=9x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply both sides by 4.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%283x%5E2-8x%2B5%29=9x%5E2 ALIGN=MIDDLE  ALT=\"4%283x%5E2-8x%2B5%29=9x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > FOIL\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12x%5E2-32x%2B20=9x%5E2 ALIGN=MIDDLE  ALT=\"12x%5E2-32x%2B20=9x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12x%5E2-32x%2B20-9x%5E2=0 ALIGN=MIDDLE  ALT=\"12x%5E2-32x%2B20-9x%5E2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract {{9x^2}}} from both sides.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2-32x%2B20=0 ALIGN=MIDDLE  ALT=\"3x%5E2-32x%2B20=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
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document.write( "Notice that the quadratic <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2-32x%2B20 ALIGN=MIDDLE  ALT=\"3x%5E2-32x%2B20\"   DISABLED_style= \"max-width:90%; height:auto;\" > is in the form of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Ax%5E2%2BBx%2BC ALIGN=MIDDLE  ALT=\"Ax%5E2%2BBx%2BC\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=3 ALIGN=MIDDLE  ALT=\"A=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B=-32 ALIGN=MIDDLE  ALT=\"B=-32\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=20 ALIGN=MIDDLE  ALT=\"C=20\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's use the quadratic formula to solve for \"x\":\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-B+%2B-+sqrt%28+B%5E2-4AC+%29%29%2F%282A%29 ALIGN=MIDDLE  ALT=\"x+=+%28-B+%2B-+sqrt%28+B%5E2-4AC+%29%29%2F%282A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the quadratic formula\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-%28-32%29+%2B-+sqrt%28+%28-32%29%5E2-4%283%29%2820%29+%29%29%2F%282%283%29%29 ALIGN=MIDDLE  ALT=\"x+=+%28-%28-32%29+%2B-+sqrt%28+%28-32%29%5E2-4%283%29%2820%29+%29%29%2F%282%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=3 ALIGN=MIDDLE  ALT=\"A=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B=-32 ALIGN=MIDDLE  ALT=\"B=-32\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=20 ALIGN=MIDDLE  ALT=\"C=20\"   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+=+%2832+%2B-+sqrt%28+%28-32%29%5E2-4%283%29%2820%29+%29%29%2F%282%283%29%29 ALIGN=MIDDLE  ALT=\"x+=+%2832+%2B-+sqrt%28+%28-32%29%5E2-4%283%29%2820%29+%29%29%2F%282%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Negate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-32 ALIGN=MIDDLE  ALT=\"-32\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=32 ALIGN=MIDDLE  ALT=\"32\"   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+=+%2832+%2B-+sqrt%28+1024-4%283%29%2820%29+%29%29%2F%282%283%29%29 ALIGN=MIDDLE  ALT=\"x+=+%2832+%2B-+sqrt%28+1024-4%283%29%2820%29+%29%29%2F%282%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-32 ALIGN=MIDDLE  ALT=\"-32\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1024 ALIGN=MIDDLE  ALT=\"1024\"   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+=+%2832+%2B-+sqrt%28+1024-240+%29%29%2F%282%283%29%29 ALIGN=MIDDLE  ALT=\"x+=+%2832+%2B-+sqrt%28+1024-240+%29%29%2F%282%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%283%29%2820%29 ALIGN=MIDDLE  ALT=\"4%283%29%2820%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=240 ALIGN=MIDDLE  ALT=\"240\"   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+=+%2832+%2B-+sqrt%28+784+%29%29%2F%282%283%29%29 ALIGN=MIDDLE  ALT=\"x+=+%2832+%2B-+sqrt%28+784+%29%29%2F%282%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=240 ALIGN=MIDDLE  ALT=\"240\"   DISABLED_style= \"max-width:90%; height:auto;\" > from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1024 ALIGN=MIDDLE  ALT=\"1024\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=784 ALIGN=MIDDLE  ALT=\"784\"   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+=+%2832+%2B-+sqrt%28+784+%29%29%2F%286%29 ALIGN=MIDDLE  ALT=\"x+=+%2832+%2B-+sqrt%28+784+%29%29%2F%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   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+=+%2832+%2B-+28%29%2F%286%29 ALIGN=MIDDLE  ALT=\"x+=+%2832+%2B-+28%29%2F%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the square root of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=784 ALIGN=MIDDLE  ALT=\"784\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=28 ALIGN=MIDDLE  ALT=\"28\"   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+=+%2832+%2B+28%29%2F%286%29 ALIGN=MIDDLE  ALT=\"x+=+%2832+%2B+28%29%2F%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2832+-+28%29%2F%286%29 ALIGN=MIDDLE  ALT=\"x+=+%2832+-+28%29%2F%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the expression. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2860%29%2F%286%29 ALIGN=MIDDLE  ALT=\"x+=+%2860%29%2F%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=++%284%29%2F%286%29 ALIGN=MIDDLE  ALT=\"x+=++%284%29%2F%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+10 ALIGN=MIDDLE  ALT=\"x+=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+2%2F3 ALIGN=MIDDLE  ALT=\"x+=+2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify. \r<BR>\n" );
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document.write( "So the <i>possible</i> solutions are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+10 ALIGN=MIDDLE  ALT=\"x+=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+2%2F3 ALIGN=MIDDLE  ALT=\"x+=+2%2F3\"   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( "However, if you plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+2%2F3 ALIGN=MIDDLE  ALT=\"x+=+2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > into the original equation, you'll get a contradiction.\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the only solution is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+10 ALIGN=MIDDLE  ALT=\"x+=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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