document.write( "Question 180304: two numbers have a difference of 16. find the numbers if the result of adding their sum and their product is a minimum. \n" ); document.write( "

Algebra.Com's Answer #135161 by josmiceli(19441) $\"\"$ $\"About$

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Call the numbers $\"x\"$ and $\"x+-+16\"$
\n" ); document.write( " $\"y+=+x+%2B+x+-+16+%2B+x%2A%28x+-+16%29\"$
\n" ); document.write( " $\"y+=+x%5E2+-16x+%2B+2x+-+16\"$
\n" ); document.write( " $\"y+=+x%5E2+-+14x+-+16\"$
\n" ); document.write( "The minimum occurs where $\"x%5Bmin%5D+=+%28-b%29%2F%282a%29\"$ and
\n" ); document.write( "the general formula is $\"y+=+ax%5E2+%2B+bx+%2B+c\"$
\n" ); document.write( " $\"a+=+1\"$
\n" ); document.write( " $\"b+=+-14\"$
\n" ); document.write( " $\"x%5Bmin%5D+=+%28-%28-14%29%29%2F%282%2A1%29\"$
\n" ); document.write( " $\"x%5Bmin%5D+=+14%2F2\"$
\n" ); document.write( " $\"x%5Bmin%5D+=+7\"$
\n" ); document.write( " $\"x%5Bmin%5D+-+16+=+-9\"$
\n" ); document.write( "The numbers are 7 and -9
\n" ); document.write( "check answer:
\n" ); document.write( "The minimum value of the sum and product is
\n" ); document.write( " $\"y+=+x%5E2+-+14x+-+16\"$
\n" ); document.write( " $\"y+=+7%5E2+-+14%2A7+-+16\"$
\n" ); document.write( " $\"y+=+49+-+98+-+16\"$
\n" ); document.write( " $\"y+=+-65\"$
\n" ); document.write( "If I say $\"x+=+6.99\"$ ,
\n" ); document.write( " $\"y+=+6.99%5E2+-+14%2A6.99+-+16\"$
\n" ); document.write( " $\"y+=+48.8601+-+97.86+-+16\"$
\n" ); document.write( " $\"y+=+-64.9999\"$ not quite a minimum
\n" ); document.write( "And if I say $\"x+=+7.01\"$
\n" ); document.write( " $\"y+=+7.01%5E2+-+14%2A7.01+-+16\"$
\n" ); document.write( " $\"y+=+49.1401+-+98.14+-+16\"$
\n" ); document.write( " $\"y+=+-64.9999\"$ not quite a minimum
\n" ); document.write( "So, $\"x+=+7\"$ really does give a minimum $\"y\"$ \n" ); document.write( "

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