Problem 60 - 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 the Graphing Calculator Basics appendix, page A2) and use ZERO or TRACE and ZOOM IN.] $$ 5 x^{2}+14 x+20=0 $$

Step-by-Step Solution

Answer

Using a graphing calculator, the solutions are approximately \(x \approx -5\) and \(x \approx -0.8\).

Step 1: Understand the Problem

We need to find the values of \(x\) that satisfy the quadratic equation \(5x^2 + 14x + 20 = 0\). The solution can be found using a graphing calculator by identifying points where the graph crosses the x-axis.

Step 2: Set Up the Graphing Calculator

Turn on your graphing calculator and set the window to the suggested range: \([-10, 10]\) for both x and y axes. This gives a broad view of the graph and ensures that important points, like roots, are visible.

Step 3: Enter the Quadratic Equation

Input the equation \(y = 5x^2 + 14x + 20\) into the graphing calculator. This plots the quadratic function on the coordinate plane.

Step 4: Use the 'Zero' Function

Utilize the 'Zero' function on the graphing calculator to find points where the graph crosses the x-axis (the roots or zeros of the function). Follow the calculator's instructions to select a point to the left and right of each crossing to calculate the zeros.

Step 5: Interpret the Results

The calculator will provide approximate values for the roots. In this case, the roots are the values of \(x\) where the quadratic graph intersects the x-axis.

Key Concepts

Graphing Calculator Techniques for Quadratic Equations

Using a graphing calculator to solve quadratic equations involves a series of strategic steps that make the process easier for students. First, the most important task is setting up your window correctly. Begin by turning on your graphing calculator and setting an appropriate window range. A recommended starting point is

• \([-10, 10]\) for both x and y axes
• This range will help in getting a broader view of the quadratic graph

After setting your window, inputting the equation is the next step. In this case, you would enter the equation as \(y = 5x^2 + 14x + 20\). This process will plot the quadratic function on your graphing calculator’s display. Once plotted, you can make use of built-in features like 'Zero' or 'Trace' to identify the points where the graph crosses the x-axis. These crossing points or intersections are essentially the roots of the quadratic equation. By carefully following the steps on your calculator to select appropriate left and right bounds, you will be able to pinpoint the roots effectively.

Root Finding Techniques with Graphing Calculators

Finding the roots of a quadratic equation graphically can be an enriching learning experience. The main task is to identify where the graph of the quadratic function intersects with the x-axis. Follow these simplified steps to make root finding straightforward:

• Start by graphing the equation as \(y = 5x^2 + 14x + 20\)
• Invoke the 'Zero' function which is typically under the CALC menu
• Select a point to the left and right of the intersection as prompted by the calculator

Once these points are selected, the graphing calculator computes the x-value that corresponds to the zero of the function, essentially the root of the equation. It’s also useful to verify by using the 'Trace' function to manually slide along the curve and identify the roots. This process enhances your understanding of the function’s behavior as it crosses the x-axis.

Understanding Quadratic Functions

Quadratic functions are polynomial equations of degree two, generally expressed in the form \(ax^2 + bx + c = 0\). In the context of quadratic graphs, they form a parabola that opens either upwards or downwards depending on the coefficient of \(x^2\).

• If \(a > 0\), the parabola opens upwards
• If \(a < 0\), it opens downwards

The roots of the quadratic function can be found where the graph crosses the x-axis. These are also known as zeros or solutions. The axis of symmetry, vertex, and the direction of opening give valuable insights into the nature of the function. Understanding these elements helps in predicting the graph's behavior and confirming the solutions obtained graphically with a calculator. This strong conceptual grasp ensures that graphical interpretations align with algebraic solutions, providing a holistic understanding of quadratic equations.