Problem 70 - College Algebra

Question

Among all of the pairs of numbers whose difference is 12, find the pair with the smallest product. What is the product?

Step-by-Step Solution

Answer

The smallest product is -36, from the pair (6, -6).

Step 1: Define the Variables

Let's define two variables: let the first number be \( x \) and the second number be \( y \). Since their difference should be 12, we can write the equation as \( x - y = 12 \).

Step 2: Express One Variable in Terms of the Other

From the equation \( x - y = 12 \), express \( y \) in terms of \( x \). This gives us the equation: \( y = x - 12 \).

Step 3: Set Up the Product Function

We want to find the pair whose product is smallest. The product \( P \) of the two numbers is \( P = x \cdot y = x(x - 12) \). This simplifies to \( P = x^2 - 12x \), which is the function we need to minimize.

Step 4: Find the Minimum of the Quadratic Function

The expression \( P = x^2 - 12x \) is a quadratic function in the standard form \( ax^2 + bx + c \) with \( a = 1 \), \( b = -12 \), and \( c = 0 \). The minimum value of this quadratic function occurs at the vertex, which has an \( x \)-coordinate of \( x = -\frac{b}{2a} = \frac{12}{2} = 6 \).

Step 5: Calculate the Corresponding Pair and Product

Substitute \( x = 6 \) back into the expression for \( y \): \( y = x - 12 = 6 - 12 = -6 \). Thus, the pair is \( (6, -6) \). The product is \( P = 6 \cdot (-6) = -36 \).

Step 6: Verify and Conclude

Verify that the difference between 6 and -6 is indeed 12, and confirm that this yields the minimum product. Therefore, the smallest product for any such pair is \(-36\).

Key Concepts

Difference of Numbers

In the given problem, we are tasked with finding pairs of numbers with a specific difference of 12. The "difference of numbers" is simply the result when one number is subtracted from another. Here, the difference we are given is 12. This means that the first number, let's call it \( x \), minus the second number, \( y \), equals 12. Mathematically, it is represented as:

• \( x - y = 12 \)

Using this equation, we can express one of the variables in terms of the other. For instance, we find \( y = x - 12 \). This relationship is crucial in transforming the problem into one involving a single variable, thus simplifying the calculations further. It allows us to explore how modifications to one number necessitate adjustments in the other to maintain the constant difference of 12.

Vertex of a Parabola

A parabola's vertex is a crucial point when dealing with quadratic functions, such as our product function \( P = x^2 - 12x \). This quadratic expression stands in the form \( ax^2 + bx + c \), where \( a = 1 \), \( b = -12 \), and \( c = 0 \). The vertex of the parabola gives the point where the function reaches its minimum or maximum value (minimum in this problem as the coefficient of \( x^2 \) is positive).

• The x-coordinate of the vertex is found using the formula:
  \(-\frac{b}{2a}\)
• Substituting our values, we have \(-\frac{-12}{2 \times 1} = 6\)

Therefore, the vertex is at \( x = 6 \). This tells us that when \( x = 6 \), the quadratic function reaches its minimum value, which in our context translates to the smallest product of the two numbers.

Product of Numbers

The concept of "product of numbers" refers to the result obtained from multiplying two numbers together. In this scenario, after determining the relationship \( y = x - 12 \), the product of the two numbers \( x \) and \( y \) is expressed as:

• \( P = x \times y = x(x - 12) = x^2 - 12x \)

This is the quadratic expression whose minimum value we want to find. By substituting the value of \( x \) from the vertex (where \( x = 6 \)), we compute \( y = 6 - 12 = -6 \). Therefore, the smallest product is:

• \( P = 6 \times (-6) = -36 \)

This negative value reflects the situation where one number is positive, and the other is negative, typical when their difference is a positive constant, yielding a negative product. Thus, the pair (6, -6) gives us the smallest product for the set conditions.