# Find two numbers whose difference is 168 and whose product is a minimum.?

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• If two numbers add up to a certain amount then the biggest product is always when the two numbers are the same.

In this case …

Half168 is 84.

So the numbers are 84 and -84

because the product is MINUS 7056 !

• Rewrite using math symbols.

Assuming largest number is x

x – y = 168 ==> y = x – 168

xy = P where P is a minimum

x(x-168) = P. This is a parabola.

We can find vertex (minimum) by putting in vertex format.

Use completing square technique

x2 – 168x = P

x^2 – 168x + 7056 = P + 7056 (add 7056 to complete square)

(x – 84)^2 = P + 7056.

Vertex occurs where x = 84. y = 84 -168 = -84.

• But if you are not in calculus:

x = one number

y = the other (higher)

y – x = 168

y = 168 + x

product = xy = x(168 + x) = 168x + x^2 = 1x^2 + 168x

If y = ax^2 + bx + c and a is positive,

a minimum occurs when x = -b / (2a)

so x = -(168) / (2 • 1) = -168/ 2 = -84

and y = 168 + x = 168 + (-84) = 84

• y = x-168

p = product = xy = x^2-168x

dP/dx = 2x -168 = 0

x = 84

y = -84

Numbers are -84 and 84

• x – y = 168

P = x * y

P = x * y

P = (168 + y) * y

P = 168 * y + y^2

dP/dy = 168 + 2y

dP/dy = 0

0 = 168 + 2y

0 = 84 + y

y = -84

x – y = 168

x – (-84) = 168

x + 84 = 168

x = 84

84 and -84

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