document.write( "Question 1183884: 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 #814382 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( " 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    as one of its roots.\r\n" );
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document.write( "The other root of the projected equation is  the sum of reciprocals   = .\r\n" );
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document.write( "In this expression, we can replace  a+b  by    and can replace  ab  by  2, since we just know it.\r\n" );
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document.write( "Making this replacement, you get   =  = .\r\n" );
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document.write( "So, the roots of the projected equation are    and  .\r\n" );
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document.write( "Hence, the sought equation is\r\n" );
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document.write( "     = 0,   or, EQUIVALENTLY,   = 0.\r\n" );
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document.write( "If you want to have the sought equation with integer coefficients, you can multiply the last equation by 8,\r\n" );
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document.write( "leaving it equivalent.\r\n" );
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document.write( "In this way, you get the ANSWER:\r\n" );
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document.write( "    the sought equation is  (2x+3)*(4x+3) = 0  with the roots  of    and  .\r\n" );
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\n" ); document.write( "\n" ); document.write( "There is another way to solve the problem.\r
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\n" ); document.write( "\n" ); document.write( "It is to find the roots of the given equation explicitly and then calculate their sum and the sum of reciprocals.\r
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\n" ); document.write( "\n" ); document.write( "It is possible, but it will lead you through the forest of calculations with radicals and fractions of complex numbers with radicals.\r
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\n" ); document.write( "\n" ); document.write( "So, this way is possible, but it is not elegant.\r
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\n" ); document.write( "\n" ); document.write( "The way which I showed you in my post, is TRULY ELEGANT, and IT IS the ONLY expected way \r
\n" ); document.write( "\n" ); document.write( "as the problem should be solved and presented.\r
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\n" ); document.write( "\n" ); document.write( "May I ask you ?\r
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document.write( "    This problem is advanced: it is destined for advanced students.\r\n" );
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document.write( "    Advanced students are THOSE who love Math and love solving tricky problems on their own.\r\n" );
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document.write( "    If so, then what is the reason for you, for an advanced student, post this problem\r\n" );
document.write( "    to the forum instead of solving it on YOUR OWN ?\r\n" );
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