document.write( "Question 1208715: Find two numbers whose difference is 22, but has the smallest possible product. \n" ); document.write( "

Algebra.Com's Answer #847118 by ikleyn(52810) $\"\"$ $\"About$

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\n" ); document.write( "Find two numbers whose difference is 22, but has the smallest possible product.
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document.write( "Let x be the greater of the two numbers; y be the smaller of the two numbers.\r\n" );
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document.write( "Then\r\n" );
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document.write( "    x - y = 22,  or  x = 22 + y.\r\n" );
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document.write( "Then the product xy is  (22+y)*y = 22y + y^2.\r\n" );
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document.write( "Complete the square\r\n" );
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document.write( "    xy = 22y + y^2 = y^2 + 22y = (y^2 + 22y + 121) - 121 = (y+11)^2 - 121.\r\n" );
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document.write( "Thus you have the expression for xy as the quadratic function in vertex form.\r\n" );
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document.write( "It shows that the minimum of xy is achieved at y= -11 and is equal to -121.\r\n" );
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document.write( "When y= -11, then  x = 22 + y = 22 + (-11) = 11.\r\n" );
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document.write( "Thus the two numbers x and y are 11 and -11, and the minimum of xy is -121.\r\n" );
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