/*! 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 45 Solve each equation by factoring... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 45

Solve each equation by factoring or the Quadratic Formula, as appropriate. $$ 4 x^{2}+24 x+40=4 $$

Short Answer

Expert verified
The solution to the equation is \( x = -3 \).

Step by step solution

01

Simplify the Equation

First, let's move all terms to one side of the equation by subtracting 4 from both sides: \[ 4x^2 + 24x + 40 - 4 = 0 \] This simplifies to: \[ 4x^2 + 24x + 36 = 0 \]
02

Factor Out the Greatest Common Factor

Notice that each term in the equation \(4x^2 + 24x + 36 = 0\) is divisible by 4, so we can factor out a 4: \[ 4(x^2 + 6x + 9) = 0 \] By factoring 4, the equation becomes: \( 4(x^2 + 6x + 9) = 0) \).Since 4 cannot be zero, we focus on \( x^2 + 6x + 9 = 0 \).
03

Factor the Quadratic

Now we need to factor the quadratic \( x^2 + 6x + 9 \). This is a perfect square trinomial, and it factors to: \[ (x + 3)^2 = 0 \]
04

Solve for x

Set the factor equal to zero and solve for \( x \): \[ (x + 3)^2 = 0 \] Take the square root of both sides: \[ x + 3 = 0 \] Thus, \( x = -3 \).

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 Quadratics
Factoring quadratics is a powerful technique used to solve quadratic equations. Quadratic equations are polynomials of degree 2, typically in the form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. By factoring, we aim to express the quadratic equation as a product of its linear factors, usually in the form \( (mx + n)(px + q) = 0 \). This allows us to apply the zero product property: if the product of two numbers is zero, then at least one of the numbers must be zero.

To factor a quadratic, follow these steps:
  • Identify the quadratic and check if the equation is set to zero.
  • Look for a greatest common factor (GCF) among the terms.
  • Try factoring by grouping if applicable.
  • If not easily factorable, consider using trial and error or the quadratic formula.
Factoring is especially efficient for simple quadratics and can sometimes be combined with other methods like completing the square.
Perfect Square Trinomial
A perfect square trinomial is a special form of a quadratic equation. It occurs when a trinomial (a polynomial with three terms) can be written as the square of a binomial. The standard form of a perfect square trinomial is \((ax)^2 + 2abx + b^2\), which factors into \((ax + b)^2\).

Recognizing perfect square trinomials can simplify solving quadratics immensely. Here's how to spot them:
  • The first and last terms are perfect squares.
  • The middle term is twice the product of the square roots of the first and last terms.
For example, in the trinomial \( x^2 + 6x + 9 \), both \( x^2 \) and \( 9 \) are perfect squares (\( x \) and \( 3 \)), and \( 6x \) is twice the product of \( x \) and \( 3 \). This forms the perfect square trinomial \((x + 3)^2\). Once transformed, solving \((x + 3)^2 = 0\) is straightforward.
Greatest Common Factor
The greatest common factor (GCF) plays a vital role in simplifying polynomial expressions before further solving. It is the largest factor that divides all terms of a polynomial evenly.

To find the GCF in quadratics like \(4x^2 + 24x + 36\), you need to:
  • Identify common numerical factors among coefficients.
  • Identify any common variables and how many times they occur in each term.
  • Factor out the GCF from the polynomial, simplifying the equation.
In our example, the terms 4, 24, and 36 are all divisible by 4. So, factoring out the GCF leads to \(4(x^2 + 6x + 9) = 0\), making the trinomial inside easier to handle. By reducing complexity, factoring the GCF can make finding solutions much more manageable.
Quadratic Formula
The quadratic formula is a universal method for solving any quadratic equation \( ax^2 + bx + c = 0 \), regardless of whether it is factorable. The formula is:\[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\] This formula uses the coefficients of the quadratic and can solve equations where factoring is difficult or impossible.

The applicability of the quadratic formula means:
  • There is no need to rely on finding factors manually.
  • The discriminant \( b^2 - 4ac \) indicates how many and what type of solutions exist.
  • It can always be used as a fallback solution method for any quadratic.
When \( b^2 - 4ac \) is positive, there are two distinct real solutions; if it's zero, there is a repeated real solution; and if negative, the solutions are complex. Although not needed for our specific example, understanding the quadratic formula is essential for mastering quadratics.

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

Simplify. $$ \frac{\left(w w^{3}\right)^{2}}{w^{3} w^{2}} $$

Simplify. $$ \frac{\left(9 x y^{3} z\right)^{2}}{3(x y z)^{2}} $$

ENVIRONMENTAL SCIENCE: Biodiversity It is well known that larger land areas can support larger numbers of species. According to one study, multiplying the land area by a factor of \(x\) multiplies the number of species by a factor of \(x^{0.239}\). Use a graphing calculator to graph \(y=x^{0.239}\). Use the window [0,100] by [0,4]. Find the multiple \(x\) for the land area that leads to triple the number of species. That is, find the value of \(x\) such that \(x^{0.239}=3\). [Hint: Either use TRACE or find where \(y_{1}=x^{0.239}\) INTERSECTs \(\left.y_{2}=3 .\right]\)

For each statement, either state that it is True (and find a property in the text that shows this) or state that it is False (and give an example to show this) $$ \left(x^{m}\right)^{n}=x^{m^{n}} $$

Electronic commerce or e-commerce, buying and selling over the Internet, has been growing rapidly. The total value of U.S. e-commerce in recent years in billions of dollars is given by the exponential function \(f(x)=226(1.11)^{x}\), where \(x\) is the number of years since \(2012 .\) Predict total e-commerce in the year 2020 .
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

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.