document.write( "Question 1183896: FInd the quadratic equation each of whose roots is the sum of a root and its reciprocal of the quadratic equation 2x^2 + 3x + 4 = 0. \n" ); document.write( "

Algebra.Com's Answer #814385 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "FInd the quadratic equation each of whose roots is the sum of a root
\n" ); document.write( "and its reciprocal of the quadratic equation 2x^2 + 3x + 4 = 0.
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\n" ); document.write( "\n" ); document.write( "            You are talking about another interpretation of the problem, previously solved.\r
\n" ); document.write( "\n" ); document.write( "            This interpretation is slightly different problem, but still very close to the previous one.\r
\n" ); document.write( "\n" ); document.write( "            It can be solved using the same method and metodology.\r
\n" ); document.write( "\n" ); document.write( "            Again, this problem is to apply the Vieta's theorem several times.\r
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document.write( "Let  \" a \"  and  \" b \"  be the roots of the given equation.\r\n" );
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document.write( "Then according to Vieta's theorem, the sum  (a+b)  is equal to   :  a + b = .\r\n" );
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document.write( "According to the same theorem, the product of the routs  ab  is equal to   = 2.\r\n" );
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document.write( "OK.  \r\n" );
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document.write( "HENCE, the projected equation has the value  A =   as one of its roots, and has the value B =   as the another root.\r\n" );
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document.write( "The sum of the roots A + B is  A + B =  =  = .\r\n" );
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document.write( "We substitute here  a+b =   and  ab = 2,  and we get  A + B =  +  =  = .\r\n" );
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document.write( "Next, A*B =  =  = 2 +  +  = 2  +  = 2  +  = \r\n" );
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document.write( "          = 2  +  = 2  +  = 2  +  = 2  -  =  -  = .\r\n" );
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document.write( "HENCE, applying the Vieta's theorem again, the projected equation has the form\r\n" );
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document.write( "    x^2 - (A+B)x + AB = 0,   or    = 0.\r\n" );
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document.write( "You can have this equation with integer coefficient by multiplying its terms by 8.  You will get then the final equation as\r\n" );
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document.write( "     = 0.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "As I said, the same methodology works.\r
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\n" ); document.write( "\n" ); document.write( "I think, that after my previous solution, an advanced student would be able to solve this one ON HIS OWN.\r
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\n" ); document.write( "\n" ); document.write( "In any case, I am glad to hear your response, which was meaningful, what RARELY HAPPENS at this forum.\r
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