document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=151472>Question 151472</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I am trying to find b^2-4ac and the number of real solutions for the equation 16-24x+9x^2=0. \r<BR>\n" );
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document.write( "If I put it in standard form ax^2 + ax + a = 0<BR>\n" );
document.write( "a=-9 b=24 c=-16\r<BR>\n" );
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document.write( "b^2-4ac= 24^2 - 4(-9)(-16)=576-576= 0\r<BR>\n" );
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document.write( "So: when the discriminat equals 0 there should be only one solution correct? But how do I find that 1 solution?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #111341 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=111341\">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=16-24x%2B9x%5E2=0 ALIGN=MIDDLE  ALT=\"16-24x%2B9x%5E2=0\"   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=9x%5E2-24x%2B16=0 ALIGN=MIDDLE  ALT=\"9x%5E2-24x%2B16=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange the terms.\r<BR>\n" );
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document.write( "From <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9x%5E2-24x%2B16 ALIGN=MIDDLE  ALT=\"9x%5E2-24x%2B16\"   DISABLED_style= \"max-width:90%; height:auto;\" > we can see that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=9 ALIGN=MIDDLE  ALT=\"a=9\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-24 ALIGN=MIDDLE  ALT=\"b=-24\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=16 ALIGN=MIDDLE  ALT=\"c=16\"   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=D=b%5E2-4ac ALIGN=MIDDLE  ALT=\"D=b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the discriminant formula.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=%28-24%29%5E2-4%289%29%2816%29 ALIGN=MIDDLE  ALT=\"D=%28-24%29%5E2-4%289%29%2816%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=9 ALIGN=MIDDLE  ALT=\"a=9\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-24 ALIGN=MIDDLE  ALT=\"b=-24\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=16 ALIGN=MIDDLE  ALT=\"c=16\"   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=D=576-4%289%29%2816%29 ALIGN=MIDDLE  ALT=\"D=576-4%289%29%2816%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-24 ALIGN=MIDDLE  ALT=\"-24\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=576 ALIGN=MIDDLE  ALT=\"576\"   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=D=576-576 ALIGN=MIDDLE  ALT=\"D=576-576\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%289%29%2816%29 ALIGN=MIDDLE  ALT=\"4%289%29%2816%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2836%29%2816%29=576 ALIGN=MIDDLE  ALT=\"%2836%29%2816%29=576\"   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=D=0 ALIGN=MIDDLE  ALT=\"D=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=576 ALIGN=MIDDLE  ALT=\"576\"   DISABLED_style= \"max-width:90%; height:auto;\" > from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=576 ALIGN=MIDDLE  ALT=\"576\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Since the discriminant is equal to zero, this means that there is one real solution.\r<BR>\n" );
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document.write( "So you are correct. However, I'm not sure where you got a=-9 b=24 c=-16 from.\r<BR>\n" );
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document.write( "To find the one solution, you have 2 options\r<BR>\n" );
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document.write( "Option # 1 Quadratic Formula (preferred method)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9x%5E2-24x%2B16=0 ALIGN=MIDDLE  ALT=\"9x%5E2-24x%2B16=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation.\r<BR>\n" );
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document.write( "Notice we have a quadratic equation 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=9 ALIGN=MIDDLE  ALT=\"a=9\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-24 ALIGN=MIDDLE  ALT=\"b=-24\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=16 ALIGN=MIDDLE  ALT=\"c=16\"   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-%28-24%29+%2B-+sqrt%28+%28-24%29%5E2-4%289%29%2816%29+%29%29%2F%282%289%29%29 ALIGN=MIDDLE  ALT=\"x+=+%28-%28-24%29+%2B-+sqrt%28+%28-24%29%5E2-4%289%29%2816%29+%29%29%2F%282%289%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=9 ALIGN=MIDDLE  ALT=\"a=9\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-24 ALIGN=MIDDLE  ALT=\"b=-24\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=16 ALIGN=MIDDLE  ALT=\"c=16\"   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+=+%2824+%2B-+sqrt%28+%28-24%29%5E2-4%289%29%2816%29+%29%29%2F%282%289%29%29 ALIGN=MIDDLE  ALT=\"x+=+%2824+%2B-+sqrt%28+%28-24%29%5E2-4%289%29%2816%29+%29%29%2F%282%289%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Negate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-24 ALIGN=MIDDLE  ALT=\"-24\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24 ALIGN=MIDDLE  ALT=\"24\"   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+=+%2824+%2B-+sqrt%28+576-4%289%29%2816%29+%29%29%2F%282%289%29%29 ALIGN=MIDDLE  ALT=\"x+=+%2824+%2B-+sqrt%28+576-4%289%29%2816%29+%29%29%2F%282%289%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-24 ALIGN=MIDDLE  ALT=\"-24\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=576 ALIGN=MIDDLE  ALT=\"576\"   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+=+%2824+%2B-+sqrt%28+576-576+%29%29%2F%282%289%29%29 ALIGN=MIDDLE  ALT=\"x+=+%2824+%2B-+sqrt%28+576-576+%29%29%2F%282%289%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%289%29%2816%29 ALIGN=MIDDLE  ALT=\"4%289%29%2816%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=576 ALIGN=MIDDLE  ALT=\"576\"   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+=+%2824+%2B-+sqrt%28+0+%29%29%2F%282%289%29%29 ALIGN=MIDDLE  ALT=\"x+=+%2824+%2B-+sqrt%28+0+%29%29%2F%282%289%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=576 ALIGN=MIDDLE  ALT=\"576\"   DISABLED_style= \"max-width:90%; height:auto;\" > from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=576 ALIGN=MIDDLE  ALT=\"576\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"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=x+=+%2824+%2B-+sqrt%28+0+%29%29%2F%2818%29 ALIGN=MIDDLE  ALT=\"x+=+%2824+%2B-+sqrt%28+0+%29%29%2F%2818%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=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18 ALIGN=MIDDLE  ALT=\"18\"   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+=+%2824+%2B-+0%29%2F%2818%29 ALIGN=MIDDLE  ALT=\"x+=+%2824+%2B-+0%29%2F%2818%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=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"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=x+=+%2824+%2B+0%29%2F%2818%29 ALIGN=MIDDLE  ALT=\"x+=+%2824+%2B+0%29%2F%2818%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2824+-+0%29%2F%2818%29 ALIGN=MIDDLE  ALT=\"x+=+%2824+-+0%29%2F%2818%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+=+%2824%29%2F%2818%29 ALIGN=MIDDLE  ALT=\"x+=+%2824%29%2F%2818%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=++%2824%29%2F%2818%29 ALIGN=MIDDLE  ALT=\"x+=++%2824%29%2F%2818%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+=+4%2F3 ALIGN=MIDDLE  ALT=\"x+=+4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+4%2F3 ALIGN=MIDDLE  ALT=\"x+=+4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify. \r<BR>\n" );
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document.write( "So our answer is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+4%2F3 ALIGN=MIDDLE  ALT=\"x+=+4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > (with a multiplicity of 2)\r<BR>\n" );
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document.write( "Option # 2 Factoring:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9x%5E2-24x%2B16=0 ALIGN=MIDDLE  ALT=\"9x%5E2-24x%2B16=0\"   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=%283x-4%29%283x-4%29=0 ALIGN=MIDDLE  ALT=\"%283x-4%29%283x-4%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor the left side (note: if you need help with factoring, check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/playground/change-this-name4450.solver>solver</a>)\r<BR>\n" );
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document.write( "Now set each factor equal to zero:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=4%2F3 ALIGN=MIDDLE  ALT=\"x=4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=4%2F3 ALIGN=MIDDLE  ALT=\"x=4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >    Now solve for x in each case\r<BR>\n" );
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document.write( "Since we have a repeating answer, our only answer is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=4%2F3 ALIGN=MIDDLE  ALT=\"x=4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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