/*! 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 111 Perform the indicated operations... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 111

Perform the indicated operations. $$ [(7 x+5)+4 y][(7 x+5)-4 y] $$

Short Answer

Expert verified
The result of the operation \((7x+5+4y)(7x+5-4y)\) is \(49x^2+70x+25 - 16y^2\).

Step by step solution

01

Determine the format of the given problem

Check if the given expression follows the pattern of difference of squares \((a+b)(a-b)\). Here \(a\) is \(7x+5\) and \(b\) is \(4y\).
02

Apply the formula of difference of squares

Replace \(a\) and \(b\) in the formula. This gives \((7x+5)^2 - (4y)^2\).
03

Expand the squared terms

Solve \((7x+5)^2\) and \((4y)^2\). This result in \(49x^2+70x+25\) and \(16y^2\) respectively.
04

Subtract the squared terms

Now subtract the second term from the first one to get the final result: \(49x^2+70x+25 - 16y^2\).

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.

Difference of Squares
One of the most useful identities in algebra is the difference of squares formula. This formula states that for any expressions \(a\) and \(b\), the product of the sum and difference, \((a+b)(a-b)\), can be expressed as \(a^2-b^2\).

By comparing the initial expression \([(7x+5)+4y][(7x+5)-4y]\) to this structure, you can see that \(a = 7x+5\) and \(b = 4y\). Recognizing this pattern allows us to simplify the problem significantly by directly applying the identity, transforming it into \((7x+5)^2 - (4y)^2\).

The key advantage here is that it reduces a complicated multiplication problem into simpler squares, making it much easier to handle.
Expanding Expressions
Expanding expressions is an essential skill in algebra, allowing you to simplify and better understand equations. Once we have identified the expression as a difference of squares, we need to expand \((7x+5)^2\) and \((4y)^2\).

Let's break it down: expanding \((7x+5)^2\) involves squaring a binomial, which requires the use of another known formula: \((a+b)^2 = a^2 + 2ab + b^2\). Applying this, we have:
  • Calculate \( (7x)^2 = 49x^2 \)
  • Calculate \( 2(7x)(5) = 70x \)
  • Calculate \( 5^2 = 25 \)
So, \((7x+5)^2 = 49x^2 + 70x + 25\).

For \((4y)^2\), the process is straightforward: simply square \(4y\), resulting in \(16y^2\). Expanding both components helps to reveal the inner parts of the expression, making subtraction easier in subsequent steps.
Algebraic Operations
Once you've expanded the terms using algebraic identities, the final step is to perform algebraic operations to obtain the solution. The primary operation in this task is subtraction.

Following our expansion, we need to subtract the result of \((4y)^2\) from \((7x+5)^2\):
  • The expression becomes \(49x^2 + 70x + 25 - 16y^2\).
By combining like and unlike terms appropriately, we simplify the problem to its final form. This step highlights the power of algebraic manipulation, as it transforms a potentially difficult problem into a simpler expression by systematically applying identities and operations.

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

Exercises will help you prepare for the material covered in the next section. Evaluate each exponential expression in $$ \frac{x^{30}}{x^{-10}} $$

Describe what it means to rationalize a denominator. Use both \(\frac{1}{\sqrt{5}}\) and \(\frac{1}{5+\sqrt{5}}\) in your explanation.

Simplify using properties of exponents. $$\left(25 x^{4} y^{6}\right)^{\frac{1}{2}}$$

Simplify by reducing the index of the radical. $$\sqrt[6]{x^{4}}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) toproduce a true statement. $$\left(4 \times 10^{3}\right)+\left(3 \times 10^{2}\right)=4.3 \times 10^{3}$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Pure 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.