Question 1027521
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If the sum of two numbers is 32 what will be the largest possible value of their product
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Let x be one of the two numbers.
Then the other number is  (32-x),  and the product of the two numbers is  {{{x*(32-x)}}} = {{{-x^2 + 32x}}}.

To answer the question, complete the square in the right side quadratic polynomial.

{{{x*(32-x)}}} = {{{-x^2 + 32x}}} = {{{-(x-16)^2 + 16^2}}}.

Now, the right side has the maximum equal to  {{{16^2}}} = {{{256}}}  at  x = 16.
For any other value of  x  the negative quadratic term in the right side of the expression makes it lesser.

So, the maximum of the product of the numbers  x  and  (32-x)  is achieved at  x = 16  and it is  256.
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