/*! 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 60 $$\text { solve each quadratic e... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 60

$$\text { solve each quadratic equation by factoring.}$$ $$3 x^{2}=12 x$$

Short Answer

Expert verified
The solutions to the quadratic equation are \(x = 0\) and \(x = 4\).

Step by step solution

01

Rewrite in Standard Form

Begin by arranging the equation into standard form which is: \(ax^2 + bx + c = 0\). This gives the equation: \(3x^2 - 12x = 0\)
02

Factorize The Equation

Factor out the greatest common factor, x, from each term on the left side of the equation. This results in: \(x(3x - 12) = 0\)
03

Apply the Zero-Product Property

According to the zero-product property, at least one of the factors must be zero for the product to be zero. This implies two solutions: \(x = 0\) and \((3x - 12) = 0\). To find the root from the second factor, we solve for x which gives: \(x = 12/3 = 4\).

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.

Factoring
Factoring is a crucial step in solving quadratic equations such as the one provided. In our specific equation, \(3x^2 = 12x\), the goal is to rewrite it in a "factored" form. This means breaking down the quadratic expression into simpler expressions (factors) multiplied together.

To factor the equation, identify the greatest common factor (GCF) shared by the terms. Here, both terms, \(3x^2\) and \(-12x\), share a common factor of \(3x\). By factoring \(3x\) out of each term, the equation simplifies to \(3x(x - 4) = 0\).
  • Tip: Always look for the GCF first. It simplifies the equation and makes further solving steps easier.

Factoring turns a complex quadratic into a product of simpler expressions, allowing us to apply other mathematical properties, such as the zero-product property, to find solutions.
Zero-Product Property
The zero-product property is a simple but powerful mathematical rule. It states that if a product of two expressions equals zero, then at least one of those expressions must also be zero. In the equation \(3x(x - 4) = 0\), you can conclude that either \(3x = 0\) or \(x - 4 = 0\).
  • For \(3x = 0\), solving for \(x\) gives \(x = 0\).
  • For \(x - 4 = 0\), solving for \(x\) results in \(x = 4\).

These solutions are derived directly from the zero-product property. It's important because it allows you to break down and easily solve for \(x\) in quadratic equations once they are factored. This property turns what could be a complex mystery of why the equation equals zero into a series of manageable steps, isolating and solving for \(x\) from each factor.
Standard Form
Before solving quadratic equations like \(3x^2 = 12x\) through factoring, it's necessary to express them in the standard form. The standard form of a quadratic equation is \(ax^2 + bx + c = 0\).

Rearranging the exercise equation, we subtract \(12x\) from both sides to achieve \(3x^2 - 12x = 0\). This step is crucial because many quadratic solving techniques, including factoring, require the equation to be set to zero.
  • Key Point: Rearread the equation to ensure all variable terms are on one side, and the constant is isolated to zero.
  • Advantage: Standard form provides a clear structure for identifying coefficients \(a\), \(b\), and \(c\) used in many solving methods.

This reformatting to standard form sets the stage for applying the zero-product property by having the equation ready for factoring or other solving strategies.

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

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$x^{3}-64=(x+4)\left(x^{2}+4 x-16\right)$$

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A city commission has proposed two tax bills. The first bill requires that a homeowner pay \(\$ 1800\) plus \(3 \%\) of the assessed home value in taxes. The second bill requires taxes of \(\$ 200\) plus \(8 \%\) of the assessed home value. What price range of home assessment would make the first bill a better deal?

a. A mathematics professor recently purchased a birthday cake for her son with the inscription $$ \text { Happy }\left(2^{\frac{5}{2}} \cdot 2^{\frac{3}{4}} \div 2^{\frac{1}{4}}\right) \text { th Birthday. } $$ How old is the son?b. The birthday boy, excited by the inscription on the cake, tried to wolf down the whole thing. Professor Mom, concerned about the possible metamorphosis of her son into a blimp, exclaimed, "Hold on! It is your birthday, so why not take \(\frac{8^{-\frac{4}{3}}+2^{-2}}{16^{-\frac{3}{4}}+2^{-1}}\) of the cake? I'll eat half of what's left over." How much of the cake did the professor eat?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. The model \(P=-0.18 n+2.1\) describes the number of pay phones, \(P,\) in millions, \(n\) years after \(2000,\) so I have to solve a linear equation to determine the number of pay phones in 2010.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I evaluated \(\frac{3 x-3}{4 x(x-1)}\) for \(x=1\) and obtained 0.
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

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.