document.write( "Question 397039: Find two numbers whose sum is 54 such that the sum of their squares is a minimum. (If a solution has a multiplicity of two, enter it in consecutive answer boxes.)
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Algebra.Com's Answer #281473 by richard1234(7193) $\"\"$ $\"About$

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We are given $\"a+%2B+b+=+54\"$ and want to minimize $\"a%5E2+%2B+b%5E2\"$ . Two ways to do this:\r
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\n" ); document.write( "\n" ); document.write( "Solution 1:
\n" ); document.write( "We could square $\"a+%2B+b+=+54\"$ to obtain $\"a%5E2+%2B+2ab+%2B+b%5E2+=+2916\"$ . We want to minimize $\"a%5E2+%2B+b%5E2\"$ and this is obtained when we maximize the value of $\"2ab\"$ . If you've ever solved problems about rectangles having fixed perimeters, and know that the maximum area occurs when the rectangle is a square (many ways to prove this) then we deduce $\"a+=+b+=+27\"$ , and $\"a%5E2+%2B+b%5E2+=+2916+-+2%2827%5E2%29+=+1458\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Solution 2:
\n" ); document.write( "By the Cauchy-Schwarz inequality,\r
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\n" ); document.write( "\n" ); document.write( " $\"%28a%5E2+%2B+b%5E2%29%281+%2B+1%29+%3E=+%28a+%2B+b%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"2%28a%5E2+%2B+b%5E2%29+%3E=+%28a+%2B+b%29%5E2+=+2916\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"a%5E2+%2B+b%5E2+%3E=+1458\"$ \r
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\n" ); document.write( "\n" ); document.write( "Thus the minimal value is 1458. This occurs when a = b = 27. \n" ); document.write( "

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