document.write( "Question 1162670: If α and β are roots of 2x² - 35x + 2 = 0. Find the value of (2α - 35)³(2β - 35)³. \n" ); document.write( "

Algebra.Com's Answer #786532 by ikleyn(52847) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "To simplify my writing/printing/typing, I will use notations \"a\" and \"b\" instead of $\"alpha\"$ and $\"beta\"$ , respectively.\r
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document.write( "    If \"a\" and \"b\" are roots of 2x² - 35x + 2 = 0, find the value of (2a - 35)³(2b - 35)³.\r\n" );
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document.write( "Since \"a\" is the root, we have  2a^2 - 35a + 2 = 0,  or  2a^2 - 35a = -2,  which is the same as\r\n" );
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document.write( "    a*(2a-35) = -2,  or  2a-35 =       (1)\r\n" );
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document.write( "Since \"b\" is the root, we have  2b^2 - 35b + 2 = 0,  or  2b^2 - 35b = -2,  which is the same as\r\n" );
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document.write( "    b*(2b-35) = -2,  or  2b-35 =       (2)\r\n" );
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document.write( "Now, since \"a\" and \"b\" are the roots of the equation 2x^2 - 35x + 2 = 0,\r\n" );
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document.write( "we have, due to the Vieta's theorem,  ab =  = 1.\r\n" );
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document.write( "Therefore,  (2a-35)*(2b-35) =  =  =  = 4.\r\n" );
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document.write( "After that, there is only one step to get the answer, and this step is to cube the last equation\r\n" );
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document.write( "     = 4^3 = 64.\r\n" );
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document.write( "ANSWER.  Under given conditions,   = 64. \r\n" );
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