Problem 61 - 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.) $$ 2 x^{2}+3 x-6=0 $$

Step-by-Step Solution

Answer

The solutions to the equation are approximately \( x = 1.09 \) and \( x = -2.75 \).

Step 1: Enter the Equation into the Graphing Calculator

Begin by setting up your graphing calculator for graphing the equation. Enter the equation \( y = 2x^2 + 3x - 6 \) into the calculator. Make sure the equation is entered correctly so that it will display accurately.

Step 2: Set the Window

Set the graphing window to an appropriate size to visualize the solutions well. For this exercise, use the window [-10, 10] for both x and y axes, which provides a broad view of the graph.

Step 3: Graph the Equation

Press the 'GRAPH' button to plot the graph of the equation. Observe the shape of the parabola, which opens upwards because the coefficient of \(x^2\) is positive.

Step 4: Use the ZERO Feature

Use the graphing calculator's ZERO feature to find the x-values where the graph crosses the x-axis. These interference points between the graph and the x-axis are the solutions to the equation.

Step 5: Find the Zeros

Follow the on-screen guidelines to move the cursor close to where the graph crosses the x-axis. Set the left and right bounds around an intersection point, and use the calculator to calculate where the graph crosses the x-axis. Repeat this for each point of intersection.

Step 6: Record the Solutions

The calculator will provide the values of x where the parabola intersects the x-axis. These are the solutions to the equation. Round each solution to two decimal places as required.

Key Concepts

Quadratic Equation

A quadratic equation is an expression that can be written in the standard form \( ax^2 + bx + c = 0 \). The graph of such an equation is called a parabola. This curve is either

• U-shaped (if \(a > 0\))
• or an upside-down U (if \(a < 0\))

depending on the coefficient of \(x^2\). In the given equation \(2x^2 + 3x - 6 = 0\), the quadratic term is \(2x^2\). Since the coefficient \(a = 2\) is positive, the graph opens upwards. You often solve quadratic equations by finding the roots, which are the x-values where

• the graph intersects the x-axis.

Using a graphing calculator helps visualize these interactions effectively.

Graphing Window

The graphing window on a calculator defines the visible area of the graph. It specifies the range for the x and y axes.
In this exercise, the recommended window is

• \([-10,10]\) for x-axis
• and \([-10,10]\) for y-axis

which provides a sufficient view to see where the graph intersects the axes. Ultimately, choosing the right window ensures you capture the key parts of a graph, especially the roots and turning points of a quadratic equation. Most graphing calculators allow you to adjust these settings quickly so that you can
easily zoom in or out to explore different regions of the graph more accurately.

ZERO Feature

The ZERO feature on a graphing calculator is handy for finding the roots of a quadratic equation. This feature identifies where the graph crosses the x-axis. These crossing points are called zeros or roots. Here's how to use it:

• First, graph the parabola.
• Next, navigate with the cursor close to where it crosses the x-axis.
• Set bounds by moving the cursor to the left and right of this intersection.
• Press 'ZERO', and the calculator provides the precise x-value.

These values are your solutions to the quadratic equation. By using this feature, you can accurately pinpoint the solutions, which are critical for successfully solving quadratic problems.