document.write( "Question 1209732: Suppose r and s are the values of x that satisfy the equation
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\n" ); document.write( "for some real number m. Find the minimum real value of (r - s)^2. \n" ); document.write( "

Algebra.Com's Answer #850043 by ikleyn(52788) $\"\"$ $\"About$

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\n" ); document.write( "Suppose r and s are the values of x that satisfy the equation
\n" ); document.write( "x^2 - 2mx + (m^2 - 6m + 11) = 0
\n" ); document.write( "for some real number m. Find the minimum real value of (r - s)^2.
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\n" ); document.write( "\n" ); document.write( "        As I read a problem, I see that it is written unprofessionally.\r
\n" ); document.write( "\n" ); document.write( "        Although from the great distance it looks OK, but in reality it should be RE-WRITTEN.\r
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\n" ); document.write( "\n" ); document.write( "        My editing is below.\r
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document.write( "          Suppose r and s are real roots of the equation\r\n" );
document.write( "              x^2 - 2mx + (m^2 - 6m + 11) = 0\r\n" );
document.write( "          for some real number m.  Find the minimum real value of (r - s)^2.\r\n" );
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\n" ); document.write( "\n" ); document.write( " Below is my solution for this edited formulation.\r
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document.write( "If you use the quadratic formula for the roots, you will see that the difference between the roots \r\n" );
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document.write( "is the square root of the discriminant\r\n" );
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document.write( "    the difference between the roots =   = .\r\n" );
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document.write( "Therefore,  (r-s)^2 =  =  = \r\n" );
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document.write( "                    =  = 24m - 44.\r\n" );
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document.write( "The roots are real numbers if and only if the discriminant  (24m - 44)  is non-negative\r\n" );
document.write( "Otherwise, the roots are complex non-real numbers.\r\n" );
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document.write( "As we consider the case when the roots r and s are real numbers,\r\n" );
document.write( "we must assume that  the number  24-44 is non-negative.\r\n" );
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document.write( "Then the difference (r-s)^2 is minimal, when the discriminant 24m-44 is zero.\r\n" );
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document.write( "It happens at m = 44/24 = 11/6.\r\n" );
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document.write( "ANSWER.  The minimum of  (r-s)^2, assuming r and s are real roots, is 0 (zero) at m = 11/6.\r\n" );
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