/*! 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 31 Solve the equation by using the ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 31

Solve the equation by using the quadratic formula where appropriate. $$x^{2}(x-1)=(x-1)^{3}$$

Short Answer

Expert verified
The solutions are \( x = 1 \) and \( x = \frac{1}{2} \).

Step by step solution

01

- Simplify the equation

First, notice that both sides of the equation can be simplified by dividing by \(x-1\), as long as \(x\eq1\). So the equation becomes: \(x^{2} = (x-1)^{2}\).
02

- Expand the right side

Expand \( (x-1)^{2} \) and set the equation: \( x^2 = x^2 - 2x + 1 \).
03

- Subtract \(x^2\) from both sides

To simplify, subtract \( x^2 \) from both sides: \( x^2 - x^2 = x^2 - 2x + 1 - x^2 \). This results in: \( 0 = -2x + 1 \).
04

- Solve for \(x\)

Rearrange the equation to solve for \( x \): \( 2x = 1 \). Divide by 2: \( x = \frac{1}{2} \).
05

- Check the omitted solution

Recall the original step where \( x-1 = 0 \) was divided out. So check \( x = 1 \) as a potential solution: \( (1)^2(1-1) = (1-1)^3 \, \ 0 = 0 \.\). Therefore, both \( x=1 \) and \( x=\frac{1}{2} \) are solutions.

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.

Quadratic Formula
The quadratic formula is a powerful tool for solving quadratic equations of the form ax^2 + bx + c = 0a. It is given by: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]The plus-minus (\( \pm \)) symbol indicates two possible solutions because the square root of a number can be positive or negative. The quadratic formula is derived from completing the square and is useful when factoring is difficult or impossible. Before using the formula, always ensure the equation is in standard form.
Simplifying Equations
Simplifying equations is a critical step in solving any algebraic problem. It involves making the equation easier to solve by combining like terms, factoring, or canceling out common factors. In the given problem, we started by dividing both sides by (x-1) to simplify the equation. It鈥檚 important to be careful when dividing by variables though. Ensure you consider cases where the denominator could be zero. This step helped reduce the problem to a simpler form and made it easier to solve.
Algebraic Expansion
Algebraic expansion involves expanding expressions that are raised to a power, such as (x-1)^2. This was necessary in our problem to make the equation easier to solve. Here鈥檚 how you expand (x-1)^2:
  • Use the distributive property (FOIL method)
  • First: Multiply the first terms (x * x = x^2)
  • Outer: Multiply the outer terms (-x)
  • Inner: Multiply the inner terms (-x)
  • Last: Multiply the last terms (1 * 1 = 1)
The expanded form of (x-1)^2 becomes x^2 - 2x + 1. Expanding expressions like this is crucial for simplifying and solving equations.
Solving Linear Equations
Solving linear equations involves isolating the variable to find its value. In the given problem, once simplified, we reached 0 = -2x + 1. To solve for x, follow these steps:
  • Move the constant term to the other side: 0 + 1 = -2x + 1 - 1i.e. 1 = -2x
  • Divide both sides by the coefficient of x: \[ x = \frac{1}{-2} \]
In this case, solving the linear equation gave x = \frac{1}{2}. Always double-check your solution by substituting it back into the original equation to ensure it satisfies the conditions.

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

Sketch the graph of the given equation. Find the intercepts; approximate to the nearest tenth where necessary. $$y=x^{2}+2 x-6$$

In Exercises \(75-84\), round your answer to the nearest tenth where necessary. One leg of a right triangle is \(9 \mathrm{cm},\) and the hypotenuse is \(30 \mathrm{cm} .\) Find the length of the other leg.

Use the method you think is the most appropriate to solve the given equation. Check your answers by using a different method. $$\frac{x+2}{x+4}=\frac{x+3}{x+6}$$

In Exercises \(1-64\), solve each of the given equations. If the equation is quadratic, use the factoring or square root method. If the equation has no real solutions, say so. $$(2 x-3)(x+4)=x(2 x+9)$$

Sketch the graph of the given equation. Find the intercepts; approximate to the nearest tenth where necessary. $$y=4 x^{2}-8 x-5$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

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.