Among all pairs of numbers whose difference is 14, find a pair whose product is as small as possible. What is the minimum product?

Verified step by step guidance

1. insert step 1> Define the two numbers as x and y, with the condition that their difference is 14, i.e., x - y = 14.
2. insert step 2> Express one variable in terms of the other using the difference condition: y = x - 14.
3. insert step 3> Write the expression for the product of the two numbers: P = x * y.
4. insert step 4> Substitute y = x - 14 into the product expression: P = x * (x - 14).
5. insert step 5> Simplify the expression to get a quadratic function: P = x^2 - 14x. To find the minimum product, determine the vertex of this parabola, which is at x = -b/(2a) for a quadratic ax^2 + bx + c.

Key Concepts

Difference of Numbers

The difference of two numbers refers to the result obtained when one number is subtracted from another. In this problem, we are given that the difference between two numbers is 14, which can be expressed mathematically as x - y = 14, where x and y are the two numbers. This relationship is crucial for setting up the equations needed to find the numbers.

Product of Numbers

The product of two numbers is the result of multiplying them together. In this context, we are tasked with minimizing the product of the two numbers, which can be represented as P = x * y. Understanding how to express the product in terms of one variable, using the difference constraint, is essential for solving the problem effectively.

Optimization

Optimization in mathematics involves finding the maximum or minimum value of a function under given constraints. In this case, we need to minimize the product of the two numbers while adhering to the condition that their difference is 14. Techniques such as substitution or calculus can be employed to find the optimal solution, which is key to determining the minimum product.