document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=188628>Question 188628</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find all real number solutions of the equation algebraically.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+sqrt%282x%2B1%29+ ALIGN=MIDDLE  ALT=\"+sqrt%282x%2B1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > - <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+sqrt%28x%2B1%29+ ALIGN=MIDDLE  ALT=\"+sqrt%28x%2B1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 2\r<BR>\n" );
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document.write( "I found that x = 0 and x = 8 but both solutions are extraneous and therefore I think the equation has no solution. Is this correct?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #141432 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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/cgi-bin/show-face.mpl?user=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=141432\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  sqrt{2x+1}-sqrt{x+1} = 2\">\r<BR>\n" );
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document.write( "Square both sides:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  (2x+1) - 2 sqrt{(2x+1)(x+1)} + (x + 1) = 4\">\r<BR>\n" );
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document.write( "Simplify:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  - 2 sqrt{(2x+1)(x+1)} = -3x + 2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  sqrt{(2x+1)(x+1)} = \frac{3x}{2} - 1\">\r<BR>\n" );
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document.write( "Square both sides:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  (2x+1)(x+1) = \frac{9x^2}{4} - 3x + 1\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  2x^2 + 3x + 1 = \frac{9x^2}{4} - 3x + 1\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  8x^2 + 12x + 4 = 9x^2 - 12x + 4\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  -x^2 + 24x = 0 \ \ \Rightarrow\ \ x^2 - 24x = 0 \ \ \Rightarrow\ \ x(x - 24) = 0\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x = 0\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  sqrt{2(0)+1}-sqrt{(0)+1} = 1 - 1 = 0 \neq 2\">\r<BR>\n" );
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document.write( "Root is extraneous\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x = 24\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  sqrt{2(24)+1}-sqrt{(24)+1} = sqrt{49} - sqrt{25} = 7 - 5 = 2\">\r<BR>\n" );
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document.write( "Answer checks.\r<BR>\n" );
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document.write( "This problem brings to mind a very interesting brain teaser.  The number 48 is very special.  If you add 1, you get 49 which is a perfect square.  If you divide it by 2 and then add 1, you get another perfect square (48/2 = 24, 24 + 1 = 25).  Find three other integers with the same property.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
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