Question 1208715: Find two numbers whose difference is 22, but has the smallest possible product.
Answer by ikleyn(52803)   (Show Source):
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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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