document.write( "Question 1131744: Find the roots of the equation x³ - 9x² + 23x - 15 = 0, if they are in AP. \n" ); document.write( "

Algebra.Com's Answer #748419 by ikleyn(52802) $\"\"$ $\"About$

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document.write( "Let a, b and c be these roots.\r\n" );
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document.write( "Since they form an AP, we can write them as  m-d, m and m+d, where m is the middle term m=b and d is the common difference,  so\r\n" );
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document.write( "    a = m-d, c = m+d.\r\n" );
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document.write( "Then according to the Vieta's theorem, the sum of the roots is equal to the coefficient at    taken with the opposite sign:\r\n" );
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document.write( "    (m-d) + m + (m+d) = 9,   or   3m = 9,  which implies  m = 9/3 = 3.\r\n" );
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document.write( "The product of the roots, using the Vieta's theorem again, is equal to the constant term taken with the opposite sign:\r\n" );
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document.write( "    (3-d)*3*(3+d) = 15,   or   = 15/3 = 5,  which implies   = 9 - 5 = 4;  hence,  d = +/- = +/-2.\r\n" );
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document.write( "In this way, the AP is  EITHER   3-2 = 1, 3, 3+2 = 5  OR  5, 3, 1,  which makes no difference.\r\n" );
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document.write( "Answer.  The roots are  1, 3 and 5.\r\n" );
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