Problem 122 - College Algebra Essentials

Question

Use a graphing utility and the graph's \(x\) -intercepts to solve each equation. Check by direct substitution. A viewing rectangle is given. $$\begin{aligned}&x^{3}+3 x^{2}-x-3=0\\\&[-6,6,1] \text { by }[-6,6,1]\end{aligned}$$

Step-by-Step Solution

Answer

The exact x-coordinates of the x-intercepts from the graph will be the solutions to the equation. They need to be confirmed by direct substitution into the original equation.

Step 1: Use Graphing Utility

First, input the equation \(x^{3}+3 x^{2}-x-3=0\) into a graphing utility. Use the viewing rectangle [-6,6,1] by [-6,6,1], which means setting the x-values and y-values to range from -6 to 6 with a scale of 1.

Step 2: Identify X-intercepts

Next, look for the points where the function intersects the x-axis.These points are called x-intercepts and they are the solutions to the equation.

Step 3: Write Down The Solutions

Record down the x-coordinates of the x-intercepts. Each x-coordinate corresponds to a solution to the equation.

Step 4: Verify Solutions By Direct Substitution

To check the validity of the found solutions, substitute each solution back into the original equation. If the left side of the equation equals to zero (0), then the solution is valid.

Key Concepts

Graphing Utility

A graphing utility is a powerful tool that allows students and mathematicians to visually analyze equations and functions. In the context of solving algebraic equations, a graphing utility can be used to plot the given equation on a coordinate system.

For instance, when dealing with a cubic equation like \(x^3 + 3x^2 - x - 3 = 0\), the graphing utility will showcase the curve that represents the equation. By visualizing the equation, it becomes much easier to understand the behavior of the function, such as where it intersects the x-axis or its overall shape—a technique especially helpful for non-linear equations where direct factoring is difficult or not possible.

X-Intercepts

The term x-intercepts refers to the points where a graph crosses the x-axis. These intercepts are of particular interest because they represent the solutions to the equation \(f(x) = 0\). In a graphical approach, identifying the x-intercepts essentially solves the equation without the need for algebraic manipulations.

When you entered the equation into a graphing utility as per the given exercise, the points at which the resulting curve intersects the x-axis are the 'roots' or solutions of the equation. Keep in mind, some equations may have multiple x-intercepts, each representing a distinct solution.

Direct Substitution

After identifying potential solutions, direct substitution is the way to confirm their accuracy. It involves taking the x-intercepts found from the graph and plugging them back into the original equation. If, after substituting, the equation balances out to zero, the solution is correct.

Verifying each solution is important because it ensures that no erroneous solutions are accepted due to potential graphing errors or misinterpretations. Throughout the practice of algebra, direct substitution is a reliable check that is often used to verify solutions obtained through graphical methods or even other algebraic techniques.

Viewing Rectangle

The viewing rectangle is the portion of the coordinate plane that is displayed by the graphing utility. The provided viewing rectangle, represented as [-6, 6, 1] by [-6, 6, 1], sets the vertical and horizontal scope for the graph.

This rectangle dictates that the graph will show x- and y-values ranging from -6 to 6, which can assist in providing a clear window to observe the behavior of the graph near the x-intercepts. A well-chosen viewing rectangle can mean the difference between an easily interpretable graph where the solutions are evident and a confusing presentation where intercepts might be missed. Choosing an appropriate viewing rectangle is part of the setup process for utilizing a graphing utility effectively.