Question

Of all numbers whose difference is 12, find the two that have the minimum product

What are the two numbers?
(Use a comma to separate answers as needed )

Answer 1

The minimum product of two numbers with a difference of 12 is obtained when those two numbers are -6 and 6, resulting in a product of -36.

Step-by-step explanation:

The problem at hand is finding two numbers whose difference is 12, and among all such number pairs, we aim to find the pair that has the minimum product. This is a typical optimization problem that can be solved through algebra.

1. Let the smaller number be x. Then the larger number is x + 12 since their difference is 12.
2. The product of these two numbers is x(x + 12). To find the minimum product, we'll need to find the minimum value of this quadratic expression.
3. To gain insight into where the minimum value lies, we can complete the square for the expression x^2 + 12x.
4. Complete the square by finding ½ of the coefficient of x (which is 6), and squaring it (which gives us 36) then add and subtract it from the expression: x^2 + 12x + 36 - 36.
5. This can be rewritten as (x + 6)^2 - 36. It's now evident that the minimum value of the expression is at x = -6.
6. Therefore, the number pair with the minimum product is -6 and 6, as -6 + 12 equals 6.
7. The minimum product is -6 × 6 = -36.

So, answering the student's question succinctly: The two numbers whose difference is 12 and have the minimum product are -6 and 6, with the product being -36.