Problem 36 - Algebra A Combined Function

Question

Solve. Find two numbers whose difference is 8 and whose product is as small as possible.

Step-by-Step Solution

Answer

The numbers are 4 and -4.

Step 1: Define the Variables

Let the first number be \( x \) and the second number be \( y \). According to the problem, \( x - y = 8 \). We will express \( y \) in terms of \( x \): \( y = x - 8 \).

Step 2: Express the Product

We want the product \( P = x \cdot y \) to be as small as possible. Substituting \( y = x - 8 \), we get the product function as \( P(x) = x(x - 8) = x^2 - 8x \).

Step 3: Find the Vertex of the Quadratic Function

The function \( P(x) = x^2 - 8x \) is a quadratic equation. The vertex form of a quadratic equation is \( P(x) = a(x-h)^2 + k \), where \( (h, k) \) is the vertex. For \( P(x) = x^2 - 8x \), the vertex \( x \) value can be found using \( h = -\frac{b}{2a} \). Here, \( a = 1 \) and \( b = -8 \). So, \( h = \frac{8}{2} = 4 \).

Step 4: Calculate the Second Number

Using the vertex \( x = 4 \), we substitute back to find \( y \). Since \( y = x - 8 \), then \( y = 4 - 8 = -4 \).

Step 5: Verify the Conditions

We have found the numbers to be 4 and -4. Their difference \( 4 - (-4) = 8 \) and their product \( 4 \times (-4) = -16 \). The conditions of the problem statement are satisfied.

Key Concepts

Vertex Form

The vertex form of a quadratic equation is a structural representation that can make analyzing certain properties of the quadratic much easier. If we have a standard form quadratic equation, such as \( ax^2 + bx + c \), the vertex form allows us to easily identify the graph's vertex. The vertex form is expressed as \( a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola.

Transposing a quadratic equation to vertex form involves completing the square, a method that restructures the equation so that it visually reveals the vertex. The formula \( h = -\frac{b}{2a} \) helps determine the x-coordinate of the vertex directly from the original equation parameters \( a \) and \( b \). For a quadratic like \( x^2 - 8x \), we identified \( h = 4 \), pinpointing the vertex at \((4, k)\), where \( k \) represents the minimum value of the product in this context.

Quadratic Function

Quadratic functions are polynomial functions with a degree of 2, ideal for modeling scenarios involving parabolic shapes. The general form is \( f(x) = ax^2 + bx + c \). Their graphs are parabolas, opening upwards if \( a > 0 \) and downwards if \( a < 0 \). These functions are particularly useful in modeling relations where one variable depends on the square of another.

In our context, the quadratic equation \( P(x) = x^2 - 8x \) arose from converting a real-world constraint about the product of two numbers. Solving quadratic functions typically involves finding the zeros, the vertex, and analyzing the function’s behavior over its domain. The vertex can either represent a maximum or minimum value, which gives us insights into optimization problems.

Product Optimization

Product optimization in mathematics involves finding the maximum or, frequently, the minimum product of two or more variables subject to given constraints. This task requires expressing the product as a function of a single variable and then identifying optimal values. In our example, the task was to find two numbers with a specified difference whose product is minimized.

The problem was approached by expressing the relationship \( y = x - 8 \), leading to the product function \( P(x) = x \cdot (x - 8) = x^2 - 8x \). The goal was to find an \( x \) that results in the smallest possible product, achieved by finding the vertex of this quadratic function. The x-coordinate of the vertex indicated the minimum point, solving for \( x = 4 \). Consequently, both numbers were determined as 4 and -4, resulting in their minimal product of -16.