document.write( "Question 263259: solve: x^5/6 + x^2/3-2x^1/2 =0\r
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Algebra.Com's Answer #854000 by ikleyn(53742) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "solve: x^5/6 + x^2/3 - 2x^1/2 = 0
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\n" ); document.write( "\n" ); document.write( "        The solution in the post by @mananth is FATALLY incorrect.\r
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\n" ); document.write( "\n" ); document.write( "        His first step is to raise all three addends in the left side of the equation to degree 6.\r
\n" ); document.write( "\n" ); document.write( "        But this step is not an equivalent transformation, so, starting from this point,\r
\n" ); document.write( "\n" ); document.write( "        his solution is inadequate.\r
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\n" ); document.write( "\n" ); document.write( "        See my correct solution below.\r
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document.write( "The original equation is \r\n" );
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document.write( "    x^5/6 + x^2/3 - 2x^1/2 = 0.      (1)\r\n" );
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document.write( "It is the same as\r\n" );
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document.write( "    x^5/6 + x^4/6 - 2x^3/6 = 0.      (2)\r\n" );
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document.write( "Factor left side, taking  x^3/6  out the parentheses as a common factor\r\n" );
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document.write( "    x^3/6*(x^2/6 + x^1/6 - 2) = 0.   (3)\r\n" );
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document.write( "One root is  x = 0,  generated by the common factor  x^3/6.\r\n" );
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document.write( "To find the roots generated by the expression in parentheses, introduce new variable  t = x^1/6.\r\n" );
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document.write( "Then the expression in parentheses is \r\n" );
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document.write( "    t^2 + r - 2.\r\n" );
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document.write( "Its roots are   =  =  =  = .\r\n" );
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document.write( "So, the roots are t = -2 and  t = 1.   \r\n" );
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document.write( "Since t = x^1/6,  we accept the positive root  t = 1  and  reject the negative root t = -2.\r\n" );
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document.write( "Thus, finally we have the solutions to the original equation  x = 0  and  x = 1.\r\n" );
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document.write( "ANSWER.  The original equation has two real solutions  x = 0  and  x = 1.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved correctly.\r
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