/*! 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 147 Factor: \(9 x^{2}-16 .\) (Sectio... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 147

Factor: \(9 x^{2}-16 .\) (Section 6.4, Example 1)

Short Answer

Expert verified
The factored form of \(9x^2 - 16\) is \((3x - 4)(3x + 4)\).

Step by step solution

01

Identify the squares

In our given problem, we need to identify the squares. The term \(9x^2\) can be written as \( (3x)^2 \), and the term -16 can be written as \( (-4)^2 \). So, our problem of \(9x^2 - 16\) can be reframed as \((3x)^2 - (4)^2\)
02

Apply difference of squares formula

We now apply the difference of squares formula \((a - b) * (a + b)\) to our problem. Here, \(a = 3x\) and \(b = 4\). Substituting these values gives us \((3x - 4) * (3x + 4)\)
03

Final answer

So, the original expression \(9x^2 - 16\) factors to \((3x - 4) * (3x + 4)\)

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.

Understanding Algebraic Expressions
Algebraic expressions are the cornerstone of algebra. They are collections of numbers, variables (like x or y), and operations (such as addition and multiplication) that represent a mathematical idea. An expression can be as simple as a single number or as complex as a multi-variable equation. For example, in the expression \(9 x^{2} - 16\), \(9 x^{2}\) and \(16\) are called terms. The term \(9 x^{2}\) consists of the variable \(x\) being squared and multiplied by 9, while \(16\) is a constant. Understanding how to manage these terms is essential when we start factoring, which is the process of breaking down an expression into simpler parts or factors.
Factoring Polynomials
Factoring polynomials is a crucial skill in intermediate algebra that simplifies expressions and solves equations. Think of factoring as the reverse of multiplication. It involves finding two or more expressions that multiply together to give the original polynomial. For the polynomial \(9 x^{2} - 16\), factoring means breaking it into two binomials that would multiply to form the given expression. This process can reveal useful information about the polynomial's properties, such as its zeros or its graph's intercept points.
Difference of Squares Formula
The difference of squares is a specific type of factoring that arises when you have a subtraction of two perfect squares, just like in the example \(9 x^{2} - 16\). The formula for factoring a difference of squares is \(a^2 - b^2 = (a - b)(a + b)\), where \(a\) and \(b\) are any expressions. This powerful tool allows you to factor an expression quickly by recognizing that both terms are squares and are being subtracted. Using this knowledge, we can transform \(9x^2 - 16\) into \( (3x)^2 - (4)^2\) which is a classic setup for the difference of squares.
Navigating Intermediate Algebra
Intermediate algebra can sometimes seem like a maze of equations and formulas, but understanding its concepts allows us to solve complex problems by applying systematic methods. It covers a broad range of topics, including factoring polynomials, functions, quadratic equations, and rational expressions. Factoring a difference of squares, as we did with \(9x^2 - 16\), is just one of the many techniques learned in this branch of algebra. The goal is to simplify expressions and solve equations, which ultimately builds a strong foundation for higher-level math and various applications in science and engineering.

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

An explosion causes debris to rise vertically with an initial speed of 72 feet per second. The formula $$h=-16 t^{2}+72 t$$ describes the height of the debris above the ground, h, in feet, t seconds after the explosion. Use this information to solve. How long will it take for the debris to hit the ground?

In Exercises \(142-146,\) use the \([\mathrm{GRAPH}]\) or \([\text { TABLE }]\) feature of a graphing utility to determine if the polynomial on the left side of each equation has been correctly factored. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$4 x^{2}-12 x+9=(4 x-3)^{2} ;[-5,5,1] \text { by }[0,20,1]$$

Use factoring to solve quadratic equation. Check by substitution or by using a graphing utility and identifying \(x\)-intercepts. \(64 w^{2}=48 w-9\)

In Exercises \(142-146,\) use the \([\mathrm{GRAPH}]\) or \([\text { TABLE }]\) feature of a graphing utility to determine if the polynomial on the left side of each equation has been correctly factored. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$\begin{aligned} &x^{4}-16=\left(x^{2}+4\right)(x+2)(x-2) ;[-5,5,1] \text { by }\\\ &[-20,20,2] \end{aligned}$$

Use factoring to solve quadratic equation. Check by substitution or by using a graphing utility and identifying \(x\)-intercepts. \(25 x^{2}=49\)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Mechanics Maths

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.