document.write( "Question 1150869: Hi\r
\n" ); document.write( "\n" ); document.write( "Could you please solve the following \r
\n" ); document.write( "\n" ); document.write( "4 to the power of x plus 6 to the power of x equals 9 to the power of x. \r
\n" ); document.write( "\n" ); document.write( "What is the value of x. \r
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Algebra.Com's Answer #772417 by ikleyn(52777) $\"\"$ $\"About$

You can put this solution on YOUR website!
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document.write( "The given exponential equation\r\n" );
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document.write( "     +  =              (1)\r\n" );
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document.write( "is equivalent to\r\n" );
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document.write( "     +  = ,   or\r\n" );
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document.write( "     +  = .          (2) \r\n" );
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document.write( "Introduce new variables  u = 2^x,  v = 3^x.  Then equation (2) takes the form\r\n" );
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document.write( "    u^2 + u*v = .\r\n" );
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document.write( "Divide both sides by u^2.  You will get\r\n" );
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document.write( "     -  - 1 = 0                (3)\r\n" );
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document.write( "Let z = .   Then equation (3) takes the form\r\n" );
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document.write( "    z^2 - z - 1 = 0.\r\n" );
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document.write( "Solve this quadratic equation using the quadratic formula\r\n" );
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document.write( "     =  = .\r\n" );
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document.write( "The roots are  \r\n" );
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document.write( "     = ,   and\r\n" );
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document.write( "     = .\r\n" );
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document.write( "Thus, we should consider two cases.\r\n" );
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document.write( "(a)   = .\r\n" );
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document.write( "     It means   =  = .\r\n" );
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document.write( "     Next, take any logarithm, log base 10, or natural logarithm \"ln\"  from both sides to continue\r\n" );
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document.write( "        = ,\r\n" );
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document.write( "        x =  /  =  = 1.1868  (approximately).\r\n" );
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document.write( "    Thus this case is completed.\r\n" );
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document.write( "(b)   = - \r\n" );
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document.write( "     It means  -  =  = .\r\n" );
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document.write( "     \r\n" );
document.write( "     The left side is negative, while the right side is positive.\r\n" );
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document.write( "     So, this case has no solutions.\r\n" );
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document.write( "ANSWER.  The original equation has only one root  x =  /  =  = 1.1868  (approximately).\r\n" );
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\n" ); document.write( "\n" ); document.write( "On solving exponential equations see also the lesson\r
\n" ); document.write( "\n" ); document.write( " - Solving exponential equations\r
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