document.write( "Question 1209732: 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. \n" ); document.write( "
Algebra.Com's Answer #850044 by mccravyedwin(407)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "The answer will be 0, the smallest possible value of (r-s)2\r\n" );
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document.write( "if and only if we can find m such that the vertex of the parabola\r\n" );
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document.write( "\"f%28x%29=x%5E2+-+2mx+%2B+%28m%5E2+-+6m+%2B+11%29+\"\r\n" );
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document.write( "is on the x-axis, where both x-intercepts, r and s, the zeros of f(x),\r\n" );
document.write( "coincide and have difference 0.\r\n" );
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document.write( "We use the vertex formula\r\n" );
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document.write( "\"%28matrix%281%2C3%2C+-b%2F%282a%29%2C+%22%2C%22%2C+f%28-b%2F%282a%29%29%5E%22%22%29%29%29\"\r\n" );
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document.write( "\"-b%2F%282a%29+=+-%28-2m%29%2F%282%281%29%29=m\"\r\n" );
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document.write( "So we need for the vertex to be the point (m, 0)\r\n" );
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document.write( "\"f%28-b%2F%282a%29%29+=+m%5E2+-+2m%28m%29+%2B+%28m%5E2+-+6m+%2B+11%29+=+-6m%2B11\"\r\n" );
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document.write( "The vertex will be on the x-axis if and only if \r\n" );
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document.write( "-6m + 11 = 0 or m = 11/6.\r\n" );
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document.write( "Since this is possible, the answer is 0.\r\n" );
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document.write( "Edwin
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