SOLUTION: Find two numbers whose difference is 22, but has the smallest possible product.

Question 1208715: Find two numbers whose difference is 22, but has the smallest possible product.
Answer by ikleyn(52803) (Show Source): You can put this solution on YOUR website!
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Find two numbers whose difference is 22, but has the smallest possible product.
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Let x be the greater of the two numbers; y be the smaller of the two numbers.

Then

    x - y = 22,  or  x = 22 + y.


Then the product xy is  (22+y)*y = 22y + y^2.


Complete the square

    xy = 22y + y^2 = y^2 + 22y = (y^2 + 22y + 121) - 121 = (y+11)^2 - 121.


Thus you have the expression for xy as the quadratic function in vertex form.


It shows that the minimum of xy is achieved at y= -11 and is equal to -121.

When y= -11, then  x = 22 + y = 22 + (-11) = 11.


Thus the two numbers x and y are 11 and -11, and the minimum of xy is -121.

Solved, with explanations.

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