document.write( "Question 1011381: Find all values of K such that equation 2^x-k(2^-x)=1 has exact one real solution \n" ); document.write( "

Algebra.Com's Answer #627029 by ikleyn(52790) $\"\"$ $\"About$

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\n" ); document.write( "Find all values of k such that equation 2^x-k(2^-x)=1 has $\"highlight%28cross%28exact%29%29\"$ exactly one real solution.
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document.write( " = .      (1)\r\n" );
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document.write( "Introduce new variable y = . Then the equation takes the form\r\n" );
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document.write( "y - k* = .\r\n" );
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document.write( "Multiply both sides by y. You will get\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( "The condition that this equation has only one root is vanishing the discriminant, i.e. \r\n" );
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document.write( "d = 0,\r\n" );
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document.write( "where d =  =  = .\r\n" );
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document.write( "It means that 1 + 4k = 0,   or   k = .\r\n" );
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document.write( "Then the equation (2) become \r\n" );
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document.write( " = .         (3)\r\n" );
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document.write( "The discriminant of this equation is zero due to selection of k. (You can check it).\r\n" );
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document.write( "The unique root of this equation is y = .\r\n" );
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document.write( "Thus the equation (1) has the unique root if and only if k = . \r\n" );
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document.write( "This root is x = -1.\r\n" );
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