document.write( "Question 1091157: Please, help me solve this problem : What is the sum of the squares of the roots of (4x^2 - 9)^4 - 10(4x^2 - 9)^2 + 9 = 0. \n" ); document.write( "

Algebra.Com's Answer #705567 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( " Please, help me solve this problem : What is the sum of the squares of the roots of (4x^2 - 9)^4 - 10(4x^2 - 9)^2 + 9 = 0.
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\n" ); document.write( "\n" ); document.write( "This problem is above the average school math.\r
\n" ); document.write( "\n" ); document.write( "It is the level of Math circles or local Math olympiads.\r
\n" ); document.write( "\n" ); document.write( "So, I will assume that your level corresponds to it.\r
\n" ); document.write( "\n" ); document.write( "Therefore, I will give you the general idea only, leaving implementation to you.\r
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document.write( "Let me start noticing that the polynomial is of degree 8 and has 8 roots that are real or/and complex numbers.\r\n" );
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document.write( "But do not be worried or scared.\r\n" );
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document.write( "First of all, there is NO NEED to solve the given equation and to find the roots explicitly.\r\n" );
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document.write( "Use the Vieta's theorem instead.\r\n" );
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document.write( "The Vietas's theorem says: \r\n" );
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document.write( "    1) the sum of all the roots  \r\n" );
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document.write( "        = \r\n" );
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document.write( "       is equal to the coefficient at  taken with the opposite sign and divided by the coefficient at .\r\n" );
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document.write( "       Now notice that the coefficient at  of the given polynomial is equal to 0 (to zero).\r\n" );
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document.write( "       Since I assumed about your level, I will not explain \"WHY\".  It is OBVIOUS.\r\n" );
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document.write( "       So,  = 0.\r\n" );
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document.write( "    2)  The sum of all the products of all pairs of the roots  is equal to the coefficient at   \r\n" );
document.write( "        divided by the leading coefficient of the polynomial.\r\n" );
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document.write( "        And you need to calculate the coefficient at  and to find   in this way.\r\n" );
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document.write( "Thanks to Vieta, we know now everything that is needed to get the answer.\r\n" );
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document.write( "Indeed,  the sum of the squares of the roots  \r\n" );
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document.write( "          =  =  = .\r\n" );
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\n" ); document.write( "\n" ); document.write( "This is the way.\r
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\n" ); document.write( "\n" ); document.write( "At this point, I just completed my role as a tutor, by providing all the needed instructions to you.\r
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\n" ); document.write( "\n" ); document.write( "The rest is on you.
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