document.write( "Question 1209940: The function f has the following properties:
\n" ); document.write( "* f(x) is defined for x > 0
\n" ); document.write( "* f(x) > 0 for all x > 0
\n" ); document.write( "* f(x - y) = \sqrt{f(xy) + 4x + 4y + 8} for all x > y > 0\r
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Algebra.Com's Answer #851137 by ikleyn(52794)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "The function f has the following properties:
\n" ); document.write( "* f(x) is defined for x > 0
\n" ); document.write( "* f(x) > 0 for all x > 0
\n" ); document.write( "* f(x - y) = \"sqrt%28f%28xy%29+%2B+4x+%2B+4y+%2B+8%29\" for all x > y > 0.\r
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\n" ); document.write( "\n" ); document.write( "        In his post,  Edwin introduced right and nice idea to work with numbers  x  and  y  such that\r
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\n" ); document.write( "\n" ); document.write( "                x-y = 1,   x*y = 1.\r
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\n" ); document.write( "\n" ); document.write( "        It leads directly to the solution,  but on the way,  in the implementation process, \r
\n" ); document.write( "\n" ); document.write( "        Edwin made arithmetic errors that lead him to wrong answer.\r
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\n" ); document.write( "\n" ); document.write( "        In this my post, I copied  Edwin's calculations,  but fixed them to repair that error.\r
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\n" ); document.write( "\n" ); document.write( "        See my solution below.   At the end,  I checked my answer to prove its validity.\r
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document.write( "\"f%28x-y%29+=+sqrt%28f%28x%2Ay%29%2B4%2Ax%2B4%2Ay%2B8%29\"   <<<---===  notice that Edwin mistakenly writes \"-8\" in this place\r\n" );
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document.write( "\"f%28x-y%29+=+sqrt%28f%28x%2Ay%29%2B4%2Ax%2B4%2Ay%2B8%29\"   <<<---===  notice that Edwin mistakenly writes \"-8\" in this place\r\n" );
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document.write( "If we can find a case of x and y where \"f%28x-y%29=f%28x%2Ay%29=f%281%29\", then we could\r\n" );
document.write( "solve for f(1).  That would require  \r\n" );
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document.write( "\"x-y=x%2Ay=1\"\r\n" );
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document.write( "\"y=1%2Fx\", \"x-1%2Fx=1\"\r\n" );
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document.write( "\"x-1%2Fx=1\"\r\n" );
document.write( "\"x%5E2-1=x\"\r\n" );
document.write( "\"x%5E2-x-1=0\"\r\n" );
document.write( "\"x=%281+%2B-+sqrt%281%2B4%29%29%2F2\" <--we can only use the + sign\r\n" );
document.write( "\"x=%281%2Bsqrt%285%29%29%2F2\"\r\n" );
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document.write( "Incidentally, that, or its reciprocal, is the golden ratio, famous in \r\n" );
document.write( "historical architecture. Let's call it \"G=%281%2Bsqrt%285%29%29%2F2\".\r\n" );
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document.write( "\"G=%281%2Bsqrt%285%29%29%2F2\" and \"G-1%2FG=1\"\r\n" );
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document.write( "\"f%28G-1%2FG%29+=+sqrt%28f%28G%2A%281%2FG%29%29%2B4%2AG%2B4%2Aexpr%281%2FG%29%2B8%29\" <<<---===  notice that Edwin mistakenly lost  \"+8\"  at this place.\r\n" );
document.write( "                                                 This place was the Edwin's fatal error.\r\n" );
document.write( "                                                 After fixing it, my numbers and my calculations are different from that by Edwin.\r\n" );
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document.write( "f(1) = \"sqrt%28f%281%29%2B4%2A%28G%2B1%2FG%29%2B8%29\"\r\n" );
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document.write( "Since \"G-1%2FG=1\", \"1%2FG=G-1\"\r\n" );
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document.write( "f(1) = sqrt(f(1) + 4*(G+G-1) + 8)}}}\r\n" );
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document.write( "f(1) = sqrt(f(1) + 4*(2G-1) + 8)}}}\r\n" );
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document.write( "f(1) = sqrt(f(1) + 8G + 4)}}}\r\n" );
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document.write( "(f(1))^2 = f(1) + 8G + 4\r\n" );
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document.write( "(f(1))^2 - f(1) - (8G + 4) = 0\r\n" );
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document.write( "Solve for f(1) using the quadratic formula\r\n" );
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document.write( "f(1) = \"%281+%2B-+sqrt%281%2B4%288G%2B4%29%29%29%2F2\"  <--- we can only use +\r\n" );
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document.write( "f(1) = \"%281+%2B+sqrt%281%2B32G%2B16%29%29%2F2\"\r\n" );
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document.write( "f(1) = \"%281+%2B+sqrt%2832G%2B17%29%29%2F2\"\r\n" );
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document.write( "Substituting  G = \"%281%2Bsqrt%285%29%29%2F2\", and doing some algebra,\r\n" );
document.write( "it simplifies to\r\n" );
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document.write( "f(1) = \"%281%2Bsqrt%2833%2B16sqrt%285%29%29%29%2F2\"\r\n" );
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document.write( "ANSWER.  f(1) = 4.646597631  approximately.\r\n" );
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document.write( "                      CHECK\r\n" );
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document.write( "I will check that the original equation is correct\r\n" );
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document.write( "    f(1) = \"sqrt%28f%281%29%2B4%2Ax%2B4%2Ay%2B8%29\".    (*)\r\n" );
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document.write( "Here left side  f(1)  is 4.646597631;\r\n" );
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document.write( "    x = \"%281%2Bsqrt%285%29%29%2F2\" = 1.618033989;  \r\n" );
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document.write( "    y = \"1%2Fx\" = \"1%2F1.618033989\" = 0.618033989  (the same as y = x - 1);\r\n" );
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document.write( "so, the right side of (*) is\r\n" );
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document.write( "    \"sqrt%284.646597631+%2B+4%2A1.618033989+%2B+4%2A0.618033989%2B8%29\" = 4.646597631,\r\n" );
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document.write( "which is PRECISELY the same as the left side.\r\n" );
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document.write( "So, my answer is correct and is confirmed.   The CHECK is completed.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Do not consider the post by @CPhill seriously.\r
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\n" ); document.write( "\n" ); document.write( "This post is simply a blatant attempt to deceive a reader
\n" ); document.write( "by presenting his fake work as a solution to a mathematical problem.\r
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\n" ); document.write( "\n" ); document.write( "It does not contain a proper mathematical deducing from the beginning to the end.\r
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\n" ); document.write( "\n" ); document.write( "I am writing this in order to express / (to describe) the true state of affairs in right words.\r
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\n" ); document.write( "\n" ); document.write( "                Regarding the post by @CPhill . . . \r
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\n" ); document.write( "\n" ); document.write( "Keep in mind that @CPhill is a pseudonym for the Google artificial intelligence.\r
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\n" ); document.write( "\n" ); document.write( "The artificial intelligence is like a baby now. It is in the experimental stage
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\n" ); document.write( "\n" ); document.write( "                It has no feeling of shame - it is shameless.\r
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\n" ); document.write( "\n" ); document.write( "This time, again,  it made an error.\r
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\n" ); document.write( "\n" ); document.write( "Although the @CPhill' solution are copy-paste  Google  AI solutions,  there is one essential difference.\r
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\n" ); document.write( "\n" ); document.write( "Every time,  Google  AI  makes a note at the end of its solutions that  Google  AI  is experimental
\n" ); document.write( "and can make errors/mistakes.\r
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\n" ); document.write( "\n" ); document.write( "All @CPhill' solutions are copy-paste of  Google  AI  solutions, with one difference:
\n" ); document.write( "@PChill never makes this notice and never says that his solutions are copy-past that of Google.
\n" ); document.write( "So, he NEVER SAYS TRUTH.\r
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\n" ); document.write( "\n" ); document.write( "Every time,  @CPhill embarrassed to tell the truth.\r
\n" ); document.write( "\n" ); document.write( "But I am not embarrassing to tell the truth,  as it is my duty at this forum.\r
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\n" ); document.write( "\n" ); document.write( "And the last my comment.\r
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\n" ); document.write( "\n" ); document.write( "When you obtain such posts from @CPhill,  remember,  that  NOBODY  is responsible for their correctness,
\n" ); document.write( "until the specialists and experts will check and confirm their correctness.\r
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\n" ); document.write( "\n" ); document.write( "Without it,  their reliability is  ZERO and their creadability is  ZERO,  too.\r
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