document.write( "Question 1197055: one solution of z3+ A z^2 -z-14 =0 is z=-2-root 3 i where A is a real number. find A and other two solutions \n" ); document.write( "

Algebra.Com's Answer #830144 by ikleyn(52832) $\"\"$ $\"About$

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\n" ); document.write( "One solution of z3+ A*z^2 - z - 14 = 0 is z = -2-root(3)*i, where A is a real number.
\n" ); document.write( "Find A and other two solutions
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\n" ); document.write( "\n" ); document.write( "            The solution in the post by math_tutor2020 may scare a reader - so complicated it is\r
\n" ); document.write( "\n" ); document.write( "            with tons of calculations.\r
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\n" ); document.write( "\n" ); document.write( "            Meanwhile, the problem can be solved mentally in a very simple way, if to use Vieta's theorem.\r
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document.write( "Since the given equation is with real coefficients (given), its complex roots are conjugated complex numbers.\r\n" );
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document.write( "Since one such root is   =   (given), the other root is   = .\r\n" );
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document.write( "Vieta's theorem says that the product of three roots of the given equation equals to the constant term \r\n" );
document.write( "with the opposite sign. So\r\n" );
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document.write( "     = 14.\r\n" );
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document.write( "The product of the two complex conjugated numbers    and    is \r\n" );
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document.write( "     = 4 + 3 = 7.\r\n" );
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document.write( "Therefore, the third root of the given equation is   =  = 2.\r\n" );
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document.write( "Now the coefficient A equals to the sum of the three roots with the opposite sign, due to the same Vieta's theorem\r\n" );
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document.write( "    A =  =  = -(-2-2+2) = 2.\r\n" );
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document.write( "ANSWER.  A = 2.  The other two roots are  and 2.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "Actually, the solution and the calculations are so simple that can be made mentally in the head.\r
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\n" ); document.write( "\n" ); document.write( "The problems of this type are usually INTENDED, DESIGNED and EXPECTED to be solved in this way, using Vieta's theorem.\r
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\n" ); document.write( "\n" ); document.write( "Then the solution is easy, elegant and provides fun for a student, showing a beauty of Math,\r
\n" ); document.write( "\n" ); document.write( "instead of making many tons of unnecessary complicated calculations.\r
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\n" ); document.write( "\n" ); document.write( " And only in this context it has an educational value.\r
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\n" ); document.write( "\n" ); document.write( "On Vieta's theorem see, for example, this Wikipedia article\r
\n" ); document.write( "\n" ); document.write( "https://en.wikipedia.org/wiki/Vieta%27s_formulas\r
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\n" ); document.write( "\n" ); document.write( "This problem is a typical of a high school Math circle level.\r
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