document.write( "Question 626330: Suppose that A, B, and C are positive constants and that x+y= C. Show that the minimum value of $\"+Ax%5E2%2BBy%5E2+\"$ occurs when $\"+x=+BC%2F%28A%2BB%29+\"$ and $\"+y=AC%2F%28A%2BB%29+\"$ \n" ); document.write( "

Algebra.Com's Answer #394115 by Edwin McCravy(20060) $\"\"$ $\"About$

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Suppose that A, B, and C are positive constants and that x + y = C. Show that the minimum value of Ax² + By² occurs when x = $\"BC%2F%28A%2BB%29\"$ and y = $\"AC%2F%28A%2BB%29+\"$
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document.write( "We know that if A > 0, the minimum value of y = Ax² + Bx + C \r\n" );
document.write( "occurs when x = \r\n" );
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document.write( "To avoid conflict of letters we re-write that as\r\n" );
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document.write( "We know that if P > 0, the minimum value of y = Px² + Qx + R \r\n" );
document.write( "occurs when x = \r\n" );
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document.write( "Since x + y = C, y = C - x, so\r\n" );
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document.write( "Minimum value of Ax² + By² = \r\n" );
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document.write( "Minimum value of Ax² + B(C - x)² =\r\n" );
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document.write( "Minimum value of Ax² + B(C - x)(C - x) =\r\n" );
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document.write( "Minimum value of Ax² + B(C² - 2Cx + x²) =\r\n" );
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document.write( "Minimum value of Ax² + BC² - 2BCx + Bx² =\r\n" );
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document.write( "Minimum value of Ax² + Bx² - 2BCx + BC² =\r\n" );
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document.write( "Minimum value of (A + B)x² - 2BCx + BC² \r\n" );
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document.write( "And by the rule above:\r\n" );
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document.write( "We know that if P > 0, the minimum value of y = Px² + Qx + R\r\n" );
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document.write( "occurs when x = \r\n" );
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document.write( "Let P = (A + B),  Q = -2BC, and  R = BC²\r\n" );
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document.write( "Minimum value of (A + B)x² - 2BCx + BC² \r\n" );
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document.write( "occurs when x =  =  = \r\n" );
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document.write( "and since y = C - x, the value of y when x =  is\r\n" );
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document.write( "y = C -  =   -  =  \r\n" );
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document.write( "· -  =  -  =\r\n" );
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document.write( " -  =  = .\r\n" );
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document.write( "Edwin

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