document.write( "Question 1146973: if x = (a + (a²+b³)^1/2)^1/3 + (a - (a²+b³)^1/2)^1/3) then x³ + 3bx ise \n" ); document.write( "

Algebra.Com's Answer #768563 by ikleyn(52908) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "            Looking into the solution by Edwin, not every school student knows about the formula Cardano.\r
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\n" ); document.write( "\n" ); document.write( "            So, I think, more simple and more accessible solution is expected and is required.\r
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\n" ); document.write( "\n" ); document.write( "            See such a solution below. It uses nothing that goes outside of the standard school Math curriculum.\r
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\n" ); document.write( "\n" ); document.write( "            Nothing --- except combining standard knowledge in the right order :)\r
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document.write( "I will introduce new variables  u =   and  v = .\r\n" );
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document.write( "Then  \r\n" );
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document.write( "     = ,\r\n" );
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document.write( "     = ,    \r\n" );
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document.write( "    x = u + v,\r\n" );
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document.write( "     = . =  =  = ;\r\n" );
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document.write( "           therefore,   =  = -b.\r\n" );
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document.write( "    Thus we have  x = u + v,    +  = 2a,  and  u*v = -b.    (1)\r\n" );
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document.write( "Next,   =  =  +  +  +  =  + .\r\n" );
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document.write( "Substitute here from (1), and you will get\r\n" );
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document.write( "     = 2a + 3*(-b)*x = 2a - 3bx.\r\n" );
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document.write( "Therefore,\r\n" );
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document.write( "     + 3bx = 2a.         ANSWER\r\n" );
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\n" ); document.write( "\n" ); document.write( "Completed and solved, and an ELEMENTARY solution is obtained.\r
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