/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 88 Use a graphing device to find al... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 88

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

Short Answer

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

Step by step solution

01

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.
02

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.
03

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.
04

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.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

\(41-58=\) Find all zeros of the polynomial. $$ P(x)=4 x^{4}+4 x^{3}+5 x^{2}+4 x+1 $$

The Cubic Formula The quadratic formula can be used to solve any quadratic (or second-degree) equation. You may have wondered if similar formulas exist for cubic (third-degree), quartic (fourth-degree), and higher-degree equations. For the depressed cubic \(x^{3}+p x+q=0\) , Cardano (page 344 ) found the following formula for one solution: \(x=\sqrt[3]{\frac{-q}{2}+\sqrt{\frac{q^{2}}{4}+\frac{p^{3}}{27}}}+\sqrt[3]{\frac{-q}{2}-\sqrt{\frac{q^{2}}{4}+\frac{p^{3}}{27}}}\) A formula for quartic equations was discovered by the Italian mathematician Ferrari in \(1540 .\) In 1824 the Norwegian mathematician Niels Henrik Abel proved that is impossible to write a quartic formula, that is, a formula for fifth-degree equations. Finally, Galois (page 327\()\) gave a criterion for determining which equations can be solved by a formula involving radicals. (a) \(x^{3}-3 x+2=0\) (b) \(x^{3}-27 x-54=0\) (c) \(x^{3}+3 x+4=0\)

\(31-40=\) Find a polynomial with integer coefficients that satisfies the given conditions. $$ Q \text { has degree } 3, \text { and zeros }-3 \text { and } 1+i $$

By the Zeros Theorem, every \(n\) th-degree polynomial equation has exactly \(n\) solutions (including possibly some that are repeated). Some of these may be real and some may be imaginary. Use a graphing device to determine how many real and imaginary solutions each equation has. (a) \(x^{4}-2 x^{3}-11 x^{2}+12 x=0\) (b) \(x^{4}-2 x^{3}-11 x^{2}+12 x-5=0\) (c) \(x^{4}-2 x^{3}-11 x^{2}+12 x+40=0\)

\(41-58=\) Find all zeros of the polynomial. $$ P(x)=x^{4}-6 x^{3}+13 x^{2}-24 x+36 $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.