document.write( "Question 1106487: Evaluate √(7/3+√(7/9+√(7/3+√(7/9+⋯)) ) ) \n" ); document.write( "
Algebra.Com's Answer #721476 by ikleyn(52781)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "I just solved similar, but much more simple problem under the link\r
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\n" ); document.write( "\n" ); document.write( "https://www.algebra.com/algebra/homework/Angles/Angles.faq.question.1106262.html\r
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\n" ); document.write( "\n" ); document.write( "        So,  I will assume that it was YOU who requested the preceding problem  (because  \"Lightning never strikes the same place twice\").\r
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\n" ); document.write( "\n" ); document.write( "        In other words,  I will assume that you are familiar with the idea and the solution of that previous problem.\r
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\n" ); document.write( "\n" ); document.write( "        Based on it,  I will be short with this one.\r
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document.write( "Let  us consider, for brewity of writing, more general expression\r\n" );
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document.write( "\"sqrt%28a+%2B+sqrt%28b+%2B+sqrt%28a+%2B+sqrt%28b%29+%2B+ellipsis%29%29%29\" = x,\r\n" );
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document.write( "where  a = \"7%2F3\",  b = \"7%2F9\".  Then  \r\n" );
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document.write( "\"%28x%5E2-a%29%5E2+-+b\" = x.                  (It is clear, and I will not spend words to justify it . . . )\r\n" );
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document.write( "It is equivalent to\r\n" );
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document.write( "\"x%5E4+-2a%2Ax%5E2+%2B+a%5E2\" - \"b\" = x,    or\r\n" );
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document.write( "\"X%5E4+-+2a%2Ax%5E2+-+x+%2B+%28a%5E2-b%29\" = 0.\r\n" );
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document.write( "Now substitute here  a = \"7%2F3\",  b = \"7%2F9\". You will get this equation in the form\r\n" );
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document.write( "\"x%5E4+-+%2814%2F3%29%2Ax%5E2+-+x+%2B+14%2F3\" = 0,   or, multiplying all the terms by 3\r\n" );
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document.write( "\"3x%5E4+-14x%5E2+-+3x+%2B+14\" = 0.\r\n" );
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document.write( "Now I will not go into details, and simply show the plot of the last polynomial.\r\n" );
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document.write( "It clearly shows that x= 2 is the root.  And now you can check it MANUALLY  (as I did . . . )\r\n" );
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document.write( "Plot y = \"3x%5E4+-+14x%5E2+-+3x+%2B+14\"\r\n" );
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document.write( "It makes me CONVINCED that  \"sqrt%287%2F3+%2B+sqrt%287%2F9+%2B+sqrt%287%2F3+%2B+sqrt%287%2F9%29+%2B+ellipsis%29%29%29\" = 2.\r\n" );
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document.write( "Check.  \"sqrt%287%2F3+%2B+sqrt%287%2F9+%2B+sqrt%287%2F3+%2B+sqrt%287%2F9%29%29%29%29\" = 1.984 (approx.)\r\n" );
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