SOLUTION: If the sum of two numbers is 32 what will be the largest possible value of their product

Let x be one of the two numbers.
Then the other number is  (32-x),  and the product of the two numbers is   = .

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

 =  = .

Now, the right side has the maximum equal to   =   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.