/*! 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 21 Solve each equation by the zero-... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 21

Solve each equation by the zero-factor property. $$4 x^{2}-4 x+1=0$$

Short Answer

Expert verified
x = 1/2

Step by step solution

01

Identify the Quadratic Equation

The given equation is already in the form of a standard quadratic equation: 4x^2 - 4x + 1 = 0
02

Use the Zero-Factor Property

To use the zero-factor property, we need to factor the quadratic equation. Notice that this quadratic equation can be rewritten as: (2x - 1)^2 = 0
03

Solve for x

Since the square of any real number is zero only if the number itself is zero, we set the factor equal to zero: 2x - 1 = 0
04

Isolate x

Solve for x by isolating it on one side of the equation:2x - 1 = 0 2x = 1 x = 1/2

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 equations
Quadratic equations are a type of polynomial equation of degree 2. They are generally written in the form:
ax^2 + bx + c = 0
In this form, 'a', 'b', and 'c' are constants, and 'x' is the variable.

What makes quadratic equations interesting is that they can have 0, 1, or 2 real roots. These roots are the values of 'x' that make the equation true.
For example, in our exercise:

4x^2 - 4x + 1 = 0
a = 4, b = -4, and c = 1.

Quadratic equations appear in various real-life situations, such as physics problems, profit calculations, and projectile motion. Understanding how to solve them is crucial for many aspects of science and engineering.
factoring
Factoring is a method used to break down complex expressions into simpler ones, making it easier to solve equations.
In our given quadratic equation:

4x^2 - 4x + 1 = 0
We can rewrite it by noticing it fits perfectly into a square of a binomial:
(2x - 1)^2 = 0

To factor a quadratic equation, look for patterns or use methods like:
  • Finding two numbers that multiply to 鈥榓c鈥� and add to 鈥榖鈥�
  • Using special factoring formulas (e.g., the difference of squares)
  • Completing the square


Factoring reveals the roots of the equation, making it easier to solve. This step is essential for applying the zero-factor property.
solving equations
Solving equations involves finding the value(s) that make the equation true. For our quadratic equation:

(2x - 1)^2 = 0

We apply the zero-factor property, which states:

If a product of factors equals zero, at least one of the factors must be zero.
Hence, we set the factor equal to zero:

2x - 1 = 0

To solve for 'x', isolate it on one side:

2x = 1
x = 1/2

By solving the equation, we've found that 'x = 1/2' is the root.
Solving quadratic equations can sometimes be tricky, but with practice, it becomes easier.

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

Solve each rational inequality. Write each solution set in interval notation. $$\frac{10}{3+2 x} \leq 5$$

Classroom Ventilation According to the American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. (ASHRAE), a nonsmoking classroom should have a ventilation rate of \(15 \mathrm{ft}^{3}\) per min for each person in the room. (a) Write an equation that models the total ventilation \(V\) (in cubic feet per hour) necessary for a classroom with \(x\) students. (b) A common unit of ventilation is air change per hour (ach). 1 ach is equivalent to exchanging all of the air in a room every hour. If \(x\) students are in a classroom having volume \(15,000 \mathrm{ft}^{3},\) determine how many air exchanges per hour \((A)\) are necessary to keep the room properly ventilated. (c) Find the necessary number of ach ( \(A\) ) if the classroom has 40 students in it. (d) In areas like bars and lounges that allow smoking, the ventilation rate should be increased to \(50 \mathrm{ft}^{3}\) per min per person. Compared to classrooms, ventilation should be increased by what factor in heavy smoking areas?

Solve each problem. The industrial process that is used to convert methanol to gasoline is carried out at a temperature range of \(680^{\circ} \mathrm{F}\) to \(780^{\circ} \mathrm{F}\). Using \(F\) as the variable, write an absolute value inequality that corresponds to this range.

When humans breathe, carbon dioxide is emitted. In one study, the emission rates of carbon dioxide by college students were measured during both lectures and exams. The average individual rate \(R_{L}\) (in grams per hour) during a lecture class satisfied the inequality $$\left|R_{L}-26.75\right| \leq 1.42,$$ whereas during an exam the rate \(R_{E}\) satisfied the inequality $$\left|R_{E}-38.75\right| \leq 2.17.$$ Use this information in Exercises. The class had 225 students. If \(T_{L}\) and \(T_{E}\) represent the total amounts of carbon dioxide in grams emitted during a one-hour lecture and exam, respectively, write inequalities that model the ranges for \(T_{L}\) and \(T_{E}\).

Solve each equation or inequality. $$|4 x+3|>0$$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Calculus

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.