Problem 88 - Algebra and Trigonometry

Question

Use a graphing device to find all real solutions of the equation, correct to two decimal places. $$ 2 x^{3}-8 x^{2}+9 x-9=0 $$

Step-by-Step Solution

Answer

The real solutions to the equation are approximately 0.82 and 3.68.

Step 1: Understand the Problem

We are asked to find all real solutions to the cubic equation \(2x^3 - 8x^2 + 9x - 9 = 0\) using a graphing device, and to express each solution correct to two decimal places.

Step 2: Graph the Function

Input the function \(f(x) = 2x^3 - 8x^2 + 9x - 9\) into a graphing device or software. Visualize the function's graph to locate the points where it intersects the x-axis, because these points are the real solutions to the equation.

Step 3: Identify the X-Intercepts

From the graph, observe the x-values where the curve crosses the x-axis. These are the real solutions of the equation. Ensure that the intersection points are clear.

Step 4: Determine Solutions to Two Decimal Places

Use the zoom or trace feature in the graphing device to pinpoint the exact x-values of the intersection points. Record these values and round them to two decimal places for precision.

Key Concepts

Graphing Devices and Their Use in Solving Equations

Graphing devices are incredibly helpful tools for visualizing equations and understanding their behavior on a graph. These devices can be specialized calculators or computer software programs that allow for detailed graph rendering. When working on solving equations, particularly cubic equations like \( 2x^3 - 8x^2 + 9x - 9 = 0 \), these devices speed up the process by providing visual insights.

Here's how they work:

• Input the equation into the device to generate its graph.
• The graph represents the equation visually, showing the curve of the function along the axes.
• Look for the points where the graph crosses the x-axis; these are the solutions to the equation.

Graphing devices make solving complex equations more manageable by offering a visual representation, allowing you to quickly locate and verify possible real solutions.

Understanding Real Solutions in the Context of Cubic Equations

Real solutions of an equation are those that can be expressed as actual numbers on the number line. For cubic equations, this involves finding the x-values where the curve touches or crosses the x-axis. These intersections indicate the real roots of the equation, solutions where the function equals zero. In our example cubic equation, \( 2x^3 - 8x^2 + 9x - 9 = 0 \), real solutions are the specific points on the x-axis that satisfy the equation.

Here's why real solutions are important:

• They give concrete answers to problems involving physical quantities.
• Real solutions indicate equilibrium points or specific conditions in applied problems.

Understanding and identifying these solutions on a graph allows learners to bridge the gap between abstract calculations and tangible results.

Precision with Decimal Approximation

When dealing with cubic equations and their real roots, it's often necessary to express solutions with a certain degree of precision. This is where decimal approximation comes in, which ensures that calculations are not only correct but also precise to a useful degree for further computations.

The process involves:

• Using graphing devices to locate x-axis interception points with precision tools like zoom or trace.
• Recording these x-values with a decimal precision, often to two decimal places, to ensure accuracy.
• Rounding the exact values to maintain consistency in results and comparisons.

Decimal approximation enhances the usability of solutions in practical scenarios, where small differences can significantly impact outcomes, making it a crucial part of solving equations like the given cubic equation.