/*! 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 26 Factor the difference of two squ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 26

Factor the difference of two squares. $$ 36 x^{2}-9 $$

Short Answer

Expert verified
(6x-3)(6x +3)

Step by step solution

01

Identify the Difference of Squares

Recognize that the given expression is in the form of a difference of squares, which follows the pattern:
02

Write Each Term as a Square

Re-write each term as a square: For the term 36x^2, notice that 36x^2 = (6x)^2 5. For the term 9, notice that 9 = 3^2
03

Apply the Difference of Squares Formula

Use the formula a^2 - b^2 = (a - b)(a + b) . Here A = 6x and B = 3. Substitute these values into the formula: (6x)^2 -3^2= (6x- 3)(6x+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.

Understanding Factoring
Factoring is an essential skill in algebra. It involves breaking down an expression into simpler components that can be multiplied to produce the original expression. One common method of factoring is by recognizing patterns, such as the difference of squares. The difference of squares occurs when you have an expression like \(a^2 - b^2\).
This can be factored using the formula: \(a^2 - b^2 = (a - b)(a + b)\).
In our example, we start with \(36x^2 - 9\). By identifying 36x^2 and 9 as squares (since \(36x^2 = (6x)^2\) and \(9 = 3^2\)), we can apply the difference of squares formula to factor the expression.
Algebraic Expressions
Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. They don't have equality signs, unlike equations. In the expression \(36x^2 - 9\), we see a combination of constants (numbers) and variables (letters representing numbers). Recognizing patterns in algebraic expressions is key to simplifying them.
When working with algebraic expressions:
  • Look for common factors.
  • Identify patterns like the difference of squares.
  • Simplify the expression by factoring.
    Using these techniques helps in identifying how to break down and work with complex expressions easily.
In our exercise, we broke down each term \(36x^2\) and \(9\) into their square components to factor the entire expression.
Understanding Polynomial Identities
Polynomial identities are equations that are true for all values of the variable involved. They help us to transform and simplify polynomial expressions. A crucial polynomial identity in this context is the difference of squares.
For any real numbers \(a\) and \(b\), the formula is:
  • \(a^2 - b^2 = (a - b)(a + b)\)
In our exercise, \(a = 6x\) and \(b = 3\). Substituting these values into the difference of squares identity, we get:
  • \(36x^2 - 9 = (6x - 3)(6x + 3)\)
This identity helps in quickly and correctly factoring the expression without needing to perform long multiplications.
Understanding polynomial identities can significantly ease solving many algebraic expressions and equations.

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

Rationalize the numerator of each expression. Assume that all variables are positive when they appear. $$\frac{\sqrt{x-7}-1}{x-8} \quad x \neq 8$$

The period \(T\), in seconds, of a pendulum of length \(l,\) in feet, may be approximated using the formula $$T=2 \pi \sqrt{\frac{l}{32}}$$ Express your answer both as a square root and as a decimal approximation. Find the period \(T\) of a pendulum whose length is 64 feet.

Simplify each expression. Express your answer so that only positive exponents occur. Assume that the variables are positive. $$\left(x^{4} y^{8}\right)^{3 / 4}$$

Simplify each expression. Express your answer so that only positive exponents occur. Assume that the variables are positive. $$ \frac{\left(16 x^{2} y^{-1 / 3}\right)^{3 / 4}}{\left(x y^{2}\right)^{1 / 4}} $$

Simplify each expression. Express your answer so that only positive exponents occur. Assume that the variables are positive. $$x^{3 / 4} x^{1 / 3} x^{-1 / 2}$$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics 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.