Problem 76 - 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.) $$-3,2$$

Step-by-Step Solution

Answer

The values are \(a = 1\), \(b = 1\), and \(c = -6\).

Step 1: Use zero-product property in reverse

According to the zero-product property, if a quadratic equation has roots \(-3\) and \(2\), it can be expressed as \((x + 3)(x - 2) = 0\). We transform the roots back into factors of the quadratic equation.

Step 2: Expand the product

To expand \((x + 3)(x - 2)\), we apply the distributive property as follows: \[(x + 3)(x - 2) = x(x - 2) + 3(x - 2) = x^2 - 2x + 3x - 6.\] Simplifying yields \(x^2 + x - 6\).

Step 3: Identify coefficients

The expanded form \(x^2 + x - 6\) can be compared to the general quadratic format \(ax^2 + bx + c\). Here, \(a = 1\), \(b = 1\), and \(c = -6\).

Key Concepts

Zero-Product Property

The zero-product property is an essential concept in solving quadratic equations. It states that if the product of two numbers is zero, then at least one of the numbers must be zero. For quadratic equations, this means that if you can express the equation as a product of two binomials set to zero, each binomial can individually be set equal to zero to find the solution for the variable.

In practical terms, let's go through an example. Suppose we have a quadratic equation with roots (-3) and 2. Using the zero-product property backward, you would write the equation as \((x + 3)(x - 2) = 0\). This breaks the quadratic into two simpler equations: \(x + 3 = 0\) or \(x - 2 = 0\). Solving these, we get the solutions \(x = -3\) or \(x = 2\), aligned perfectly with our given root values.

This property is powerful because it allows us to solve quadratics by splitting them into more manageable parts. The key takeaways are to transform roots into factorized forms and then apply the zero-product property.

Roots of Quadratic Equations

The term "roots of a quadratic equation" refers to the values of the variable that make the equation true, i.e., complete its balance to zero. More formally, these are the solutions to the equation \(ax^2 + bx + c = 0\). Quadratic equations can often yield two roots, which may be real or complex numbers.

In our example, we work with the roots (-3) and 2. These two numbers are the solutions to the equation we ultimately expanded to \(x^2 + x - 6 = 0\). The fascinating part about roots is that they represent the x-intercepts of the graph of the quadratic function. That's to say, they are the points where the parabola, represented by the quadratic, crosses the x-axis.

Finding roots is not only about solving an equation; it's also about understanding the behavior of quadratic functions. This is vital in fields like physics and engineering, where these equations often describe parabolic motion or paths.

Coefficients of Quadratics

Coefficients in a quadratic equation \(ax^2 + bx + c\) are the numbers that multiply each term: \(a\), \(b\), and \(c\). They hold the key to describing the nature of the quadratic equation.

In the equation we derived, \(x^2 + x - 6 = 0\), the coefficients are \(a = 1\), \(b = 1\), and \(c = -6\). Each plays a unique role:

• The leading coefficient \(a\) determines the direction of the parabola's opening. If \(a\) is positive, it opens upwards; if negative, downwards.
• The coefficient \(b\) impacts the symmetry of the parabola. Together with \(a\), it influences the location of the vertex.
• The constant term \(c\) represents the y-intercept, showing where the parabola crosses the y-axis.

Understanding these coefficients is crucial because slight changes in their values can significantly alter the graph's shape and position. Therefore, mastering how to identify and interpret these coefficients will deepen your understanding of quadratic behavior and solutions.