Problem 58 - Brief Applied Calculus

Question

Solve each equation using a graphing calculator. [Hint: Begin with the window \([-10,10]\) by \([-10,10]\) or another of your choice (see Useful Hint in Graphing Calculator Terminology following the Preface) and use ZERO, SOLVE, or TRACE and ZOOM IN.] (Round answers to two decimal places.) $$ 3 x^{2}-6 x+5=2 $$

Step-by-Step Solution

Answer

x = 1.00 is the double root.

Step 1: Set Up the Equation for Graphing

First, rewrite the equation so that it equals zero: \(3x^2 - 6x + 5 - 2 = 0\). This simplifies to \(3x^2 - 6x + 3 = 0\). We will graph the equation \(y = 3x^2 - 6x + 3\) and look for the x-values where \(y = 0\).

Step 2: Enter the Equation in the Calculator

Input the equation \(y = 3x^2 - 6x + 3\) into the graphing calculator. Make sure that your calculator window is set to \([-10, 10]\) for both x and y axes, providing a broad view of the graph's behavior.

Step 3: Graph the Equation

Use the graph function to display the curve of the equation on the screen. You should see a parabola opening upwards.

Step 4: Find the Roots Using the Zero Feature

Access the ZERO feature from the calculator menu. This tool helps find where the graph crosses the x-axis, which corresponds to the solutions of the equation. Follow the calculator instructions to move to one of the intersection points (roots) on the x-axis and confirm the zero.

Step 5: Record the x-values

Repeat the zero-finding process for any other x-intercepts. Round each x-value to two decimal places as required. Based on the graph, you should find two x-values where the equation equals zero, potentially using the TRACE feature to verify the points.

Key Concepts

Quadratic Equations

Quadratic equations are an important part of algebra, consisting of polynomial equations of degree 2. These types of equations generally have the form: \[ ax^2 + bx + c = 0 \] where \( a \), \( b \), and \( c \) are constants and \( x \) represents an unknown variable. Here, \( a \) cannot be zero, because otherwise the equation would not be quadratic but linear. The graph of a quadratic equation forms a U-shaped curve known as a parabola. Understanding quadratic equations is crucial because they appear in various real-world problems, such as physics, engineering, and economics.

It's essential to know how to solve quadratic equations to fully grasp their applications. This is typically done using several methods: factoring, completing the square, using the quadratic formula, or graphing. Each of these strategies provide different insights into the nature and solutions of the equation.

Finding Roots

Finding the roots of a quadratic equation is about discovering the values of \( x \) that make the equation equal zero. In mathematical terms, these roots are the points where the graph intersects the x-axis.

• One common method to find the roots is by factoring the quadratic if possible. However, not all quadratics can be factored easily. So, other techniques like using a graphing calculator become handy.


• Setting the equation to zero is generally the starting point. For example, the equation \( 3x^2 - 6x + 3 = 0 \) in our exercise is ready for root finding.

After preparing the equations, utilizing a calculator's ZERO, SOLVE, or TRACE functions allows the user to move along the curve and identify the x-intercepts with precision. For rounded accuracy, results are typically rounded to two decimal places.

These tools not only aid in precision but also provide a visual understanding of the equation's behavior, which can reinforce the student’s comprehension.

Parabola

A parabola is a symmetrical, curved U-shaped graph that represents a quadratic equation. It's an essential figure in mathematics because its properties have wide-ranging applications.

• The direction in which a parabola opens (upwards or downwards) is determined by the sign of the coefficient \( a \) in the quadratic equation \( ax^2 + bx + c = 0 \). If \( a \) is positive, like in the equation \( 3x^2 - 6x + 3 \), the parabola opens upwards.


• The vertex of the parabola is the highest or lowest point, depending on the direction it opens. The vertex can be a helpful reference point when graphing the parabola or when identifying intercepts.

The x-intercepts of the parabola are where the graph crosses the x-axis, representing the solutions or roots of the quadratic equation. Engaging a graphing calculator to visualize the parabola helps students intuitively understand the equation by seeing the intersection points that correspond to the roots.

This graphical approach not only solves the equation but also strengthens mathematical visualization skills.