Problem 70 - College Algebra with Corequisite Support

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.

Step 1: Define Variables

Let's define two variables, \( x \) and \( y \), representing the two numbers. We are given that the difference between these numbers is 12, so we have the equation \( x - y = 12 \).

Step 2: Express One Variable in Terms of the Other

Using the equation from Step 1, we can express \( y \) in terms of \( x \). So, \( y = x - 12 \).

Step 3: Write the Product Function

The product of the two numbers is the function \( P = xy \). Substituting \( y = x - 12 \) into this equation, we get \( P = x(x - 12) = x^2 - 12x \).

Step 4: Find the Vertex of the Parabola

The expression for the product \( P = x^2 - 12x \) is a quadratic equation, which defines a parabola. To find the minimum product, locate the vertex of this parabola. The x-coordinate of the vertex of a quadratic function \( ax^2 + bx + c \) is given by \( x = -\frac{b}{2a} \). For our function, \( a = 1 \) and \( b = -12 \), so \( x = -\frac{-12}{2 \cdot 1} = 6 \).

Step 5: Calculate the Corresponding y-value

Using \( x = 6 \), find \( y \) using the equation \( y = x - 12 \). Thus, \( y = 6 - 12 = -6 \).

Step 6: Compute the Product

Now calculate the product of the numbers. Substituting \( x = 6 \) and \( y = -6 \) into the product equation \( P = xy \): \( P = 6 \times (-6) = -36 \).

Key Concepts

Parabola Vertex

In the realm of quadratic functions, the vertex of a parabola serves as a crucial point. It often represents the minimum or maximum value of the quadratic function, depending on the parabola's orientation. For a quadratic equation of the form \( ax^2 + bx + c \), the vertex \( x \)-coordinate can be computed using the formula \( x = -\frac{b}{2a} \). This formula directly gives us the point where the parabola turns. In the original exercise, the function \( P = x^2 - 12x \) describes a parabola opening upwards due to the positive coefficient of \( x^2 \). This means the vertex indicates the minimum point, which we're interested in for optimization problems involving minimization of value, such as finding the smallest product.

Optimization Problems

Optimization problems in mathematics are all about finding the best solution from a set of possible options. In our context, optimization aims to discover the smallest product of two numbers whose difference is 12. By translating the problem into a function, \( P = x(x-12) = x^2 - 12x \), we seek to minimize this expression. Solving this involves finding the vertex of the associated parabola, as the vertex gives us the minimum point for upward-opening parabolas. The minimum product occurs at this vertex, leading to the solution of such problems.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, usually in the format \( ax^2 + bx + c = 0 \). These are fundamental in algebra and often form parabolas when plotted. In problems like the one given, quadratic functions are instrumental because they allow us to apply methods such as completing the square or directly using the vertex formula for quick solutions. The equation \( P = x^2 - 12x \) is simple but rich with insights: the coefficient \( a \) dictates the parabola's direction, and using it along with \( b \), we easily find the vertex to solve the optimization question. Understanding quadratic equations is essential for tackling diverse mathematical problems effectively, especially those involving minima or maxima.