Expand The Expression: \[$(x+5)(x-5)\$\]

Feb 28, 2025 by ADMIN 41 views

**Expanding Algebraic Expressions: A Step-by-Step Guide**

Introduction

In algebra, expanding expressions is a crucial skill that helps us simplify complex equations and solve problems efficiently. In this article, we will focus on expanding the expression ${$ (x+5)(x-5)$}$, which is a fundamental concept in algebra. We will break down the process into manageable steps, making it easy to understand and apply.

What is Expanding an Expression?

Expanding an expression involves multiplying two or more terms together to simplify the equation. In the case of the given expression, we have two binomials, ${$ (x+5)$}$ and ${$ (x-5)$}$, which we need to multiply together.

Step 1: Multiply the First Terms

To expand the expression, we start by multiplying the first terms of each binomial. In this case, we multiply ${$ x$}$ by ${$ x$}$, which gives us ${$ x^2$}$.

Step 2: Multiply the Outer Terms

Next, we multiply the outer terms of each binomial. We multiply ${$ x$}$ by ${$ -5$}$, which gives us ${$ -5x$}$.

Step 3: Multiply the Inner Terms

Now, we multiply the inner terms of each binomial. We multiply ${ $5\$}$ by ${$ x$}$, which gives us ${ $5x\$}$ .

Step 4: Multiply the Last Terms

Finally, we multiply the last terms of each binomial. We multiply ${ $5\$}$ by ${$ -5$}$, which gives us ${$ -25$}$.

Step 5: Combine Like Terms

Now that we have multiplied all the terms, we need to combine like terms. In this case, we have two terms with the variable ${$ x$}$, which are ${$ -5x$}$ and ${ $5x\$}$ . We can combine these terms by adding them together, which gives us ${ $0x\$}$ or simply ${ $0\$}$ .

The Final Answer

After combining like terms, we are left with the final answer: ${$ x^2 - 25$}$.

Conclusion

Expanding algebraic expressions is a fundamental skill that helps us simplify complex equations and solve problems efficiently. By following the steps outlined in this article, we can expand the expression ${$ (x+5)(x-5)$}$ and arrive at the final answer: ${$ x^2 - 25$}$. With practice and patience, you can master the art of expanding algebraic expressions and become proficient in solving problems in mathematics.

Real-World Applications

Expanding algebraic expressions has numerous real-world applications in fields such as physics, engineering, and economics. For example, in physics, expanding expressions is used to describe the motion of objects under the influence of forces. In engineering, expanding expressions is used to design and optimize systems, such as bridges and buildings. In economics, expanding expressions is used to model and analyze economic systems, such as supply and demand.

Tips and Tricks

Here are some tips and tricks to help you expand algebraic expressions efficiently:

Use the distributive property: The distributive property states that ${$ a(b+c) = ab + ac$}$. This property can be used to expand expressions by multiplying each term inside the parentheses by the term outside the parentheses.
Combine like terms: Like terms are terms that have the same variable and exponent. Combining like terms involves adding or subtracting the coefficients of like terms.
Use algebraic identities: Algebraic identities are equations that are true for all values of the variables. For example, the identity ${$ a^2 - b^2 = (a+b)(a-b)$}$ can be used to expand expressions.

Common Mistakes

Here are some common mistakes to avoid when expanding algebraic expressions:

Forgetting to multiply terms: Make sure to multiply all the terms in the expression.
Not combining like terms: Combine like terms to simplify the expression.
Using the wrong algebraic identity: Use the correct algebraic identity to expand the expression.

Conclusion

Introduction

In our previous article, we explored the concept of expanding algebraic expressions and provided a step-by-step guide on how to expand the expression ${$ (x+5)(x-5)$}$. In this article, we will answer some frequently asked questions (FAQs) related to expanding algebraic expressions.

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that states that ${$ a(b+c) = ab + ac$}$. This property can be used to expand expressions by multiplying each term inside the parentheses by the term outside the parentheses.

Q: How do I expand an expression with multiple terms?

A: To expand an expression with multiple terms, you need to follow the order of operations (PEMDAS):

Parentheses: Evaluate any expressions inside parentheses.
Exponents: Evaluate any exponential expressions.
Multiplication and Division: Evaluate any multiplication and division operations from left to right.
Addition and Subtraction: Evaluate any addition and subtraction operations from left to right.

Q: What is the difference between expanding and factoring?

A: Expanding and factoring are two different operations in algebra. Expanding involves multiplying two or more terms together to simplify the equation, while factoring involves expressing an expression as a product of simpler expressions.

Q: How do I simplify an expression after expanding it?

A: To simplify an expression after expanding it, you need to combine like terms. Like terms are terms that have the same variable and exponent. Combining like terms involves adding or subtracting the coefficients of like terms.

Q: What are some common algebraic identities?

A: Some common algebraic identities include:

${$ a^2 - b^2 = (a+b)(a-b)$]
[$a^2 + b^2 = (a+b)^2 - 2ab$]
[$a^3 - b^3 = (a-b)(a^2 + ab + b^2)$]

Q: How do I use algebraic identities to expand expressions?

A: To use algebraic identities to expand expressions, you need to identify the type of expression you are working with and apply the corresponding identity. For example, if you have an expression of the form [ $a^2 - b^2\$}$ , you can use the identity [$a^2 - b^2 = (a+b)(a-b)$] to expand it.

Q: What are some common mistakes to avoid when expanding algebraic expressions?

A: Some common mistakes to avoid when expanding algebraic expressions include:

Forgetting to multiply terms: Make sure to multiply all the terms in the expression.
Not combining like terms: Combine like terms to simplify the expression.
Using the wrong algebraic identity: Use the correct algebraic identity to expand the expression.

Conclusion

Expanding algebraic expressions is a fundamental skill that helps us simplify complex equations and solve problems efficiently. By following the steps outlined in this article and practicing regularly, you can master the art of expanding algebraic expressions and become proficient in solving problems in mathematics.

Additional Resources

For more information on expanding algebraic expressions, check out the following resources:

Algebra textbooks: Check out algebra textbooks for more information on expanding algebraic expressions.
Online resources: Check out online resources such as Khan Academy, Mathway, and Wolfram Alpha for more information on expanding algebraic expressions.
Practice problems: Practice expanding algebraic expressions with practice problems to reinforce your understanding.

Final Tips

Here are some final tips to help you expand algebraic expressions efficiently:

Practice regularly: Practice expanding algebraic expressions regularly to reinforce your understanding.
Use algebraic identities: Use algebraic identities to simplify expressions and make them easier to expand.
Combine like terms: Combine like terms to simplify expressions and make them easier to expand.

By following these tips and practicing regularly, you can master the art of expanding algebraic expressions and become proficient in solving problems in mathematics.