document.write( "Question 1209873: Find all x such that\r
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Algebra.Com's Answer #850950 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "sqrt{3x^2 + 2x + 1} + sqrt{3x^2 + 2x - 3} = 20.
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document.write( "Let  u = ,  v =   are new variables.\r\n" );
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document.write( "Then for new variables we have this system of equations\r\n" );
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document.write( "    u + v = 20,      (1)\r\n" );
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document.write( "    u^2 - v^2 = 4    (2)\r\n" );
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document.write( "We can re-write equation (2) this way\r\n" );
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document.write( "    (u+v)*(u-v) = 4.\r\n" );
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document.write( "Now replace (u+v) by 20 in this equation, based on (1).  You will get\r\n" );
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document.write( "    20(u-v) - 4,  or  u - v = 4/20 = 0.2.\r\n" );
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document.write( "Thus the system (1), (2) takes the form\r\n" );
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document.write( "    u + v = 20,      (1')\r\n" );
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document.write( "    u - v = 0.2      (2')\r\n" );
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document.write( "The solution to (1'), (2') can be found mentally\r\n" );
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document.write( "    u = 10.1  (half the sum of 20 and 0.2)  and  v = 9.9 (half the difference of 20 and 0.2)\r\n" );
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document.write( "From here, we get with the first square root\r\n" );
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document.write( "     = 10.1,\r\n" );
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document.write( "    3x^2 + 2x + 1 =  = 102.01,\r\n" );
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document.write( "    3x^2 + 2x - 101.01 = 0,\r\n" );
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document.write( "     =  = 5.47882...,  \r\n" );
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document.write( "     =  = -6.14549...   using the quadratic formula.\r\n" );
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document.write( "With the second square root we get\r\n" );
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document.write( "     = 9.9,\r\n" );
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document.write( "    3x^2 + 2x - 3 =  = 98.01,\r\n" );
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document.write( "    3x^2 + 2x - 101.01 = 0,\r\n" );
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document.write( "    It is the same equation as above, so it does not add additional roots.\r\n" );
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document.write( "So, there are 2 approximate solutions :  -6.14549  and  5.47882.    ANSWER\r\n" );
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