/*! 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 Factor each expression completel... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 60

Factor each expression completely. $$ (a+b)^{2}-100 $$

Short Answer

Expert verified
Answer: The factored form of the expression is $((a+b) + 10)((a+b) - 10)$.

Step by step solution

01

Identify Difference of Squares

The given expression is: $$(a+b)^2 - 100$$ Notice that both terms are perfect squares: \((a+b)\) is squared and \(100 = 10^2\). This means we have a difference of squares.
02

Apply Difference of Squares Formula

The formula for factoring a difference of squares is: $$A^2 - B^2 = (A+B)(A-B)$$ In our case, \(A = (a+b)\) and \(B = 10\).
03

Substitute and Simplify

Plug the values of \(A\) and \(B\) into the difference of squares formula: $$((a+b) + 10)((a+b) - 10)$$ This is the complete factored form of the given expression.

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 Expressions
Factoring expressions is a crucial skill in algebra that involves breaking down an expression into simpler components or "factors" that, when multiplied together, return the original expression. This process is akin to reversing multiplication.

For example, if you have the expression \((x^2 - 9)\), it can be factored into \((x + 3)(x - 3)\). The factors \((x + 3)\) and \((x - 3)\) are simpler expressions. Factoring can simplify algebraic expressions, making them easier to solve or analyze.

  • Identify common factors or special patterns, such as squares or cubes.
  • Apply appropriate formulas or methods like the difference of squares, trinomial factoring, etc.
  • Check your work by multiplying the factors back to ensure they multiply to the original expression.
Practice different methods of factoring to become proficient in this essential algebraic technique.
Perfect Squares
Perfect squares are numbers or expressions that can be expressed as a product of an integer or a variable multiplied by itself. They are important in recognizing special patterns in algebraic expressions, such as the difference of squares.

Consider 9, which is a perfect square since it equals \(3^2\). Similarly, expressions like \((a+b)^2\) are perfect squares, as they are the result of squaring another expression, in this case, \((a+b)\).

  • Perfect squares are characterized by their "square rootable" nature.
  • Found in various algebraic patterns, they simplify the process of factoring and reducing expressions.
  • Recognizing perfect squares can help identify the potential use of the difference of squares formula as demonstrated in the original problem.
Remember, spotting these patterns can save you time and effort in solving algebraic equations.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operations like addition and multiplication. They are the core building blocks in algebra.

An algebraic expression can be something simple like \(x + 5\), or more complex with multiple operations and variables, such as \(3x^2 - 4xy + 7\). Solving or simplifying these expressions often involves using algebraic properties, recognizing patterns, and applying mathematical rules.

  • Involves constants (fixed numbers) and variables (symbols that represent numbers or values).
  • Can be manipulated to simplify them or solve equations when set equal to another expression.
  • Understanding the structure of these expressions helps in applying different techniques such as factoring, expanding, or substituting values.
Having a strong grasp of algebraic expressions allows you to tackle more advanced topics in algebra and beyond.

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 each expression. Assume any factors you cancel are not zero. $$ \frac{\frac{4 a b^{3}}{3}}{\frac{2 b}{a^{2}}} $$

Are the two expressions equivalent? \(5 x+100\) and \(5(x+500)\)

Factor the expressions in exercises. $$ r^{3}-14 r^{2} s^{3}+49 r s^{6} $$

Factor each expression completely. $$ 16 s^{3} t-24 s^{2} t^{2}+9 s t^{3} $$

Factor the expressions in exercises. $$ x^{2}+26 x+169 $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

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.