document.write( "Question 1104988: in a quadratic equation, a student made a mistake in copying the coefficient of x^(2) and got the roots of 2 and 3. Another student made a mistake copying the constant term and got the roots of 4 and 6. What are the correct roots?
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Algebra.Com's Answer #719731 by ikleyn(52781) $\"\"$ $\"About$

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document.write( "0.  Let the original equation be   = 0.     (1)\r\n" );
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document.write( "1.  After the 1-st student incorrectly copied the coefficient at  ,  the equation took the form\r\n" );
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document.write( "     = 0.     (2)\r\n" );
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document.write( "    Since its roots are 2 and 3, we have this decomposition\r\n" );
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document.write( "     = d*(x-2)*(x-3),     or\r\n" );
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document.write( "     = .\r\n" );
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document.write( "    So, the original equation (1) has the two lowest degree terms -5dx + 6d:\r\n" );
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document.write( "         =     (3)\r\n" );
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document.write( "    with some unknown coefficients  \"a\"  and  \"d\".\r\n" );
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document.write( "2.   After the 2-nd student incorrectly copied the constant term,  the equation took the form\r\n" );
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document.write( "     = 0.     (4)\r\n" );
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document.write( "    Since its roots are 4 and 6, we have this decomposition\r\n" );
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document.write( "     = a*(x-4)*(x-6),     or\r\n" );
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document.write( "     = .\r\n" );
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document.write( "    It implies  -5d = -10a,  which in turn  implies  d = 2a.\r\n" );
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document.write( "    Now from (3) we conclude that the original equation (polynomial) is/was\r\n" );
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document.write( "         = .\r\n" );
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document.write( "    Its roots are the same as for equation   = 0.\r\n" );
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document.write( "    And they are   =  = .\r\n" );
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document.write( "Answer.  The roots of the original equation are   =   and   = .\r\n" );
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