document.write( "Question 1184166: Solve algebraic expression $\"+a%5E6+-12a%5E4+%2B+48a%5E2+-+61+=+0+\"$ and find the real roots. \n" ); document.write( "

Algebra.Com's Answer #814701 by ikleyn(52786) $\"\"$ $\"About$

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\n" ); document.write( "Solve algebraic $\"cross%28expression%29\"$ EQUATION $\"+a%5E6+-12a%5E4+%2B+48a%5E2+-+61+=+0+\"$ and find the real roots.
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\n" ); document.write( "\n" ); document.write( " It looks as a miracle, but I will give a precise solution to the given equation.\r
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document.write( "The starting equation is\r\n" );
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document.write( "     = .       (1)\r\n" );
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document.write( "Recall the standard formula  (x-y)^3 = x^3 - 3x^2y + 3x*y^2 - y^3.\r\n" );
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document.write( "In the formula, put  x = a^2,  y = 4.  You will get\r\n" );
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document.write( "     =  + 3.\r\n" );
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document.write( "Hence, the given equation is EQUIVALENT to\r\n" );
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document.write( "     = -3.      (2)\r\n" );
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document.write( "It implies that     \r\n" );
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document.write( "     = ,\r\n" );
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document.write( "or\r\n" );
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document.write( "     = - .    (3)\r\n" );
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document.write( "Hence,  \r\n" );
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document.write( "     = ,\r\n" );
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document.write( "    a = +/- .     (4)\r\n" );
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document.write( "Approximate numerical values of the roots (4) are  +/- 1.599297.\r\n" );
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