Expand And, If Necessary, Combine Like Terms.${ (2x + 5)(2x - 5) = \square }$

Mar 13, 2025 by ADMIN 79 views

Introduction

When it comes to solving algebraic expressions, one of the most fundamental concepts is expanding and combining like terms. In this article, we will delve into the world of algebra and explore the process of expanding and combining like terms, using the given expression (2x + 5)(2x - 5) as a prime example.

What are Like Terms?

Before we dive into the process of expanding and combining like terms, it's essential to understand what like terms are. Like terms are algebraic expressions that have the same variable(s) raised to the same power. In other words, they are terms that have the same combination of variables and exponents.

For example, 2x and 4x are like terms because they both have the variable x raised to the power of 1. Similarly, 3y^2 and 2y^2 are like terms because they both have the variable y raised to the power of 2.

Expanding the Expression

Now that we have a solid understanding of like terms, let's focus on expanding the given expression (2x + 5)(2x - 5). To expand this expression, we need to multiply each term in the first parentheses by each term in the second parentheses.

Using the distributive property, we can multiply each term in the first parentheses by each term in the second parentheses as follows:

(2x + 5)(2x - 5) = 2x(2x) + 2x(-5) + 5(2x) + 5(-5)

Simplifying the Expression

Now that we have expanded the expression, let's simplify it by combining like terms. To do this, we need to combine the terms that have the same variable(s) raised to the same power.

Using the distributive property, we can simplify the expression as follows:

2x(2x) = 4x^2 2x(-5) = -10x 5(2x) = 10x 5(-5) = -25

Now, let's combine the like terms:

4x^2 - 10x + 10x - 25

Combining Like Terms

Now that we have simplified the expression, let's combine the like terms. To do this, we need to combine the terms that have the same variable(s) raised to the same power.

Using the distributive property, we can combine the like terms as follows:

4x^2 - 10x + 10x - 25 = 4x^2 - 25

Conclusion

In this article, we used the expression (2x + 5)(2x - 5) as a prime example of how to expand and combine like terms. We started by expanding the expression using the distributive property, and then simplified it by combining like terms.

Tips and Tricks

Here are some tips and tricks to help you expand and combine like terms like a pro:

Use the distributive property: The distributive property is a fundamental concept in algebra that allows us to multiply each term in one parentheses by each term in the other parentheses.
Identify like terms: Like terms are algebraic expressions that have the same variable(s) raised to the same power. Identifying like terms is essential to combining them.
Combine like terms: Combining like terms is a simple process that involves adding or subtracting the coefficients of the like terms.
Simplify the expression: Simplifying the expression is the final step in expanding and combining like terms. It involves combining like terms and eliminating any unnecessary terms.

Practice Problems

Here are some practice problems to help you expand and combine like terms:

Expand and simplify the expression (3x + 2)(2x - 3)
Expand and simplify the expression (x + 4)(x - 2)
Expand and simplify the expression (2x - 5)(x + 3)

Real-World Applications

Expanding and combining like terms has numerous real-world applications in fields such as physics, engineering, and economics. Here are some examples:

Physics: In physics, expanding and combining like terms is used to simplify complex equations that describe the motion of objects.
Engineering: In engineering, expanding and combining like terms is used to simplify complex equations that describe the behavior of electrical circuits.
Economics: In economics, expanding and combining like terms is used to simplify complex equations that describe the behavior of economic systems.

Conclusion

Whether you're a student or a professional, expanding and combining like terms is a skill that will serve you well in your future endeavors. So, practice it regularly and become a master of algebra!

Introduction

In our previous article, we explored the concept of expanding and combining like terms in algebra. We used the expression (2x + 5)(2x - 5) as a prime example of how to expand and simplify expressions. In this article, we will answer some frequently asked questions about expanding and combining like terms.

Q: What are like terms?

A: Like terms are algebraic expressions that have the same variable(s) raised to the same power. In other words, they are terms that have the same combination of variables and exponents.

Q: How do I identify like terms?

A: To identify like terms, you need to look for terms that have the same variable(s) raised to the same power. For example, 2x and 4x are like terms because they both have the variable x raised to the power of 1.

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that allows us to multiply each term in one parentheses by each term in the other parentheses. It is used to expand expressions and simplify them.

Q: How do I expand an expression using the distributive property?

A: To expand an expression using the distributive property, you need to multiply each term in one parentheses by each term in the other parentheses. For example, to expand the expression (2x + 5)(2x - 5), you would multiply each term in the first parentheses by each term in the second parentheses.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the like terms. For example, to combine the terms 2x and 4x, you would add their coefficients, resulting in 6x.

Q: What is the difference between expanding and combining like terms?

A: Expanding an expression involves multiplying each term in one parentheses by each term in the other parentheses, while combining like terms involves adding or subtracting the coefficients of the like terms.

Q: Why is it important to expand and combine like terms?

A: Expanding and combining like terms is important because it allows us to simplify complex expressions and make them easier to work with. It is a fundamental concept in algebra and is used in a wide range of applications, from physics and engineering to economics and finance.

Q: Can you give me some examples of how to expand and combine like terms?

A: Here are some examples:

Expand and simplify the expression (3x + 2)(2x - 3)
Expand and simplify the expression (x + 4)(x - 2)
Expand and simplify the expression (2x - 5)(x + 3)

Q: What are some real-world applications of expanding and combining like terms?

A: Expanding and combining like terms has numerous real-world applications in fields such as physics, engineering, and economics. Here are some examples:

In physics, expanding and combining like terms is used to simplify complex equations that describe the motion of objects.
In engineering, expanding and combining like terms is used to simplify complex equations that describe the behavior of electrical circuits.
In economics, expanding and combining like terms is used to simplify complex equations that describe the behavior of economic systems.

Q: How can I practice expanding and combining like terms?

A: There are many ways to practice expanding and combining like terms, including:

Working through practice problems and exercises
Using online resources and tools, such as algebra calculators and worksheets
Asking a teacher or tutor for help and guidance
Joining a study group or online community to work with others and get feedback

Q: What are some common mistakes to avoid when expanding and combining like terms?

A: Here are some common mistakes to avoid when expanding and combining like terms:

Failing to identify like terms
Failing to use the distributive property
Failing to combine like terms correctly
Making errors when multiplying or dividing terms

Conclusion

In conclusion, expanding and combining like terms is a fundamental concept in algebra that allows us to simplify complex expressions. By understanding what like terms are and using the distributive property, we can expand and simplify expressions with ease. We hope that this Q&A article has been helpful in answering some of your questions about expanding and combining like terms.