/*! 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 29 Find each product. $$ (5 y+3... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 29

Find each product. $$ (5 y+3 x)(5 y-3 x) $$

Short Answer

Expert verified
25y^2 - 9x^2

Step by step solution

01

Identify the Binomials

The given expression is in the form of (a + b)(a - b). Here, identify 5y as 'a' and 3x as 'b'.
02

Use the Difference of Squares Formula

Apply the difference of squares formula: (a + b)(a - b) = a^2 - b^2. Substitute 'a' = 5y and 'b' = 3x into the formula.
03

Substitute and Calculate

Substitute into the formula: (5y)^2 - (3x)^2. Calculate each term: (5y)^2 = 25y^2 and (3x)^2 = 9x^2.
04

Write the Final Answer

Subtract the two squared terms: 25y^2 - 9x^2. This is the simplified product.

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
The difference of squares is a special algebraic identity. It is used to simplify the product of two binomials in the form (a + b)(a - b). The result will be the difference between the squares of a and b. This means: (a + b)(a - b) = a^2 - b^2. For example, in the equation (5y + 3x)(5y - 3x), we identify 5y as 'a' and 3x as 'b'. Then, we use the formula: (5y)^2 - (3x)^2. This simplifies to 25y^2 - 9x^2. The difference of squares helps in getting the final product quickly.
Binomial Multiplication
Binomial multiplication involves multiplying two binomials. A binomial has two terms, such as (a + b) or (a - b). Multiplying (a + b) by (a - b) results in four partial products. However, using the difference of squares identity, the process becomes simpler. Instead of calculating each partial product separately, we use the identity directly to find the result: a^2 - b^2. This makes multiplication easier and faster for expressions like (5y + 3x)(5y - 3x), resulting in 25y^2 - 9x^2.
Simplifying Expressions
Simplifying expressions means making them easier to understand and work with. For quadratic expressions like (5y + 3x)(5y - 3x), we use algebraic identities such as the difference of squares to simplify them. Following the identity (a + b)(a - b) = a^2 - b^2, substitute 5y for 'a' and 3x for 'b'. We then get (5y)^2 - (3x)^2, simplifying further to 25y^2 - 9x^2. Using these methods helps in solving problems efficiently.
Quadratic Expressions
Quadratic expressions involve terms where the variables are squared, like 25y^2 - 9x^2. These expressions often arise from multiplying binomials. When using the formula for the difference of squares, our result is already a quadratic expression. For instance, from the binomials (5y + 3x)(5y - 3x), applying 25y^2 - 9x^2, we see each term is squared. Recognizing these patterns and using the right formulas makes working with quadratic expressions simpler and more intuitive.

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

Evaluate. $$ 49 \div 10,000 $$

Perform each division using the "long division" process. $$ \frac{2 r^{3}-5 r^{2}-6 r+15}{r-3} $$

Find each product. \(-5 r(r+1)^{3}\)

Perform each division. $$ \frac{y^{3}+1}{y+1} $$

If it costs \(\left(4 x^{5}+3 x^{4}+2 x^{3}+9 x^{2}-29 x+2\right)\) dollars to fertilize a garden, and fertilizer costs \((x+2)\) dollars per square yard, write an expression, in square yards, for the area of the garden.
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Applied Mathematics

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.