Problem 56 - 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.) $$ 3 x^{2}+18=15 x $$

Step-by-Step Solution

Answer

Solutions are approximately \(x \approx 2\) and \(x \approx 3\).

Step 1: Rewrite the Equation

First, rearrange the given equation into standard quadratic form. We start with \(3x^2 + 18 = 15x\). Subtract \(15x\) from both sides to obtain: \[3x^2 - 15x + 18 = 0\]

Step 2: Set Up the Graphing Calculator

Open your graphing calculator and enter the equation \(y = 3x^2 - 15x + 18\). Ensure that your graphing window is set to \([-10, 10]\) for both x and y axis, so that you have a clear view of where the parabola intersects the x-axis.

Step 3: Graph the Equation

With the equation entered, graph the function. Observe where the parabola intersects the x-axis. These intersection points are the solutions to the equation.

Step 4: Use the ZERO Feature

Utilize the ZERO feature on the calculator to find the x-values where the graph crosses the x-axis (i.e., where \(y = 0\)). Position the cursor near one intersection point and follow the calculator prompts to identify the zero. Record this x-value.

Step 5: Trace or Zoom for Better Accuracy

If needed, use the TRACE feature or ZOOM IN function to get a closer view and more precisely determine the zeros (where the graph intersects with the x-axis). Repeat from Step 4 to accurately find all zeros.

Step 6: Record the Solutions

Round the x-values determined from your graphing calculator to two decimal places. These values are the solutions to your original equation.

Step 7: Confirm the Solutions

Verify if the solutions satisfy the original equation by substituting them back into \(3x^2 + 18 = 15x\). Check if both sides of the equation are equal.

Key Concepts

Graphing Calculator

A graphing calculator is a powerful tool used in solving and visualizing mathematical problems, including quadratic equations. It helps to easily identify the shape and behavior of graphs over a defined range. In this exercise, you begin by setting up the graphing calculator window to

• Display a range for both x and y from \([-10, 10]\).
• Enter the equation of the quadratic function, such as \(y = 3x^2 - 15x + 18\).

The visual representation is crucial for identifying crucial points like the intersection with the x-axis, which is essential in solving the quadratic equation. Using graphing calculators can also save time, reduce manual error, and serve as a verification tool for algebraic solutions. Don't forget to familiarize yourself with functions like ZERO, SOLVE, or TRACE, which allow you to navigate the graph and pinpoint necessary coordinates.

Quadratic Equation Solutions

A quadratic equation is typically expressed in its standard form as \(ax^2 + bx + c = 0\). The challenge lies in finding the values of x, known as "solutions" or "roots", that satisfy the equation. The graphical method provides an intuitive way of determining these roots. You do so by following these easy steps:

• Graph the quadratic equation on a calculator.
• Locate points where the parabola intersects the x-axis, which represent the solutions.
• Use the ZERO function to get precise values for these intersection points.

It's essential to rearrange the equation to align with the graphing calculator's input format, making sure the right and left sides are equal to zero. After graphing, intersection points are found by looking where the graph dips or rises to cross the x-axis. These intersections are the solutions and often need to be rounded for simplicity, usually to two decimal places.

Graphical Method for Solving Equations

Solving quadratic equations graphically involves visual representation to find approximate solutions where algebraic methods may be cumbersome. By graphing the quadratic equation, you:

• Gain an understanding of the shape and direction of the parabola.
• Identify visually where it crosses the x-axis, representing potential solutions.
• Utilize features like TRACE or ZOOM IN to refine accuracy in reading coordinates.

The graphical approach not only aids in finding solutions but also in analyzing the graph's symmetries, vertex, and other properties. While using TRACE or ZOOM IN functions, you're able to examine any specific part of the graph for finer details. This method is particularly helpful for double-checking solutions obtained from algebraic processes and understanding how changes in coefficients affect the graph's shape and intersection points.