Problem 75 - A Graphical Approach to College Algebra

Question

Concept Check For each pair of numbers, find the values of \(a, b,\) and \(c\) for which the quadratic equation ax \(^{2}+b x+c=0\) has the given numbers as solutions. Answers may vary. (Hint: Use the zero-product property in reverse.) $$4,5$$

Step-by-Step Solution

Answer

The coefficients are \(a = 1\), \(b = -9\), and \(c = 20\).

Step 1: Understanding the Problem

We need to find the coefficients \(a, b,\) and \(c\) such that the quadratic equation \(ax^2 + bx + c = 0\) has the solutions \(x = 4\) and \(x = 5\).

Step 2: Set Up the Equation Using the Solutions

According to the zero-product property, if \(4\) and \(5\) are solutions, the quadratic equation can be expressed in factored form as \((x - 4)(x - 5) = 0\).

Step 3: Expand the Factored Equation

Expand \((x - 4)(x - 5)\). Start by using the distributive property: \(x \cdot x - 5 \cdot x - 4 \cdot x + 20 = x^2 - 9x + 20\).

Step 4: Determine the Coefficients

Compare the expanded quadratic expression \(x^2 - 9x + 20\) to the general form \(ax^2 + bx + c\). From this, we can identify \(a = 1\), \(b = -9\), and \(c = 20\).

Key Concepts

Zero-Product Property

The zero-product property is a simple yet powerful concept in algebra. It states that if the product of two factors is zero, then at least one of the factors must be zero. In mathematical terms, if \( a \cdot b = 0 \), then either \( a = 0 \), \( b = 0 \), or both. This property is particularly useful when dealing with quadratic equations because it allows us to determine the points at which the equation equals zero.

To apply this in solving equations like \( (x - 4)(x - 5) = 0 \), we set each factor equal to zero to find the solutions of the quadratic equation. Thus, \( x - 4 = 0 \) or \( x - 5 = 0 \) gives us the solutions \( x = 4 \) and \( x = 5 \). This technique is a cornerstone in breaking down and solving quadratic equations efficiently.

Factored Form

Factored form is an expression of a polynomial equation as a product of its linear factors. For a quadratic equation, transforming it into its factored form simplifies the process of finding the roots. The factored form of a quadratic equation \( ax^2 + bx + c = 0 \) is typically represented as \( (x - p)(x - q) = 0 \), where \( p \) and \( q \) are the solutions of the equation.

In our exercise, because the solutions are given as 4 and 5, the factored form becomes \( (x - 4)(x - 5) = 0 \). This representation not only helps in identifying the roots quickly but also reflects the zero-product property. To solve a quadratic equation efficiently, it is often advantageous to first convert it to its factored form if possible. This step leverages both the structure and the properties of polynomials to streamline the problem-solving process.

Quadratic Coefficients

Quadratic coefficients are the numerical values that multiply the variable terms in a standard quadratic equation of the form \( ax^2 + bx + c = 0 \). In this equation, \( a \), \( b \), and \( c \) are the quadratic coefficients. They play a crucial role in defining the properties of the quadratic graph, such as its direction, width, and vertical position.

Let's consider our solution where the factored form \((x - 4)(x - 5)\) expands to \(x^2 - 9x + 20\). Comparing this to \(ax^2 + bx + c\), we determine the coefficients:

• \( a = 1 \): This means the parabola opens upwards and has standard width.
• \( b = -9 \): This affects the symmetry and the vertex of the parabola along the x-axis.
• \( c = 20 \): This is the y-intercept, where the parabola crosses the y-axis.

Understanding and manipulating these coefficients is key to mastering quadratic equations and analyzing their behavior.