(5𝑥 + 11)(5𝑥 − 11), Expand It And Mention The Identity Used Please

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Expanding the Algebraic Expression (5𝑥 + 11)(5𝑥 − 11)

The given algebraic expression is a product of two binomials, (5𝑥 + 11) and (5𝑥 − 11). To expand this expression, we will use the distributive property, which states that for any real numbers a, b, and c, a(b + c) = ab + ac.

Step 1: Apply the Distributive Property

To expand the given expression, we will apply the distributive property to each term in the first binomial (5𝑥 + 11) and multiply it with each term in the second binomial (5𝑥 − 11).

(5𝑥 + 11)(5𝑥 − 11) = (5𝑥)(5𝑥) + (5𝑥)(−11) + (11)(5𝑥) + (11)(−11)

Step 2: Simplify the Expression

Now, let's simplify each term in the expression.

(5𝑥)(5𝑥) = 25𝑥² (5𝑥)(−11) = −55𝑥 (11)(5𝑥) = 55𝑥 (11)(−11) = −121

Step 3: Combine Like Terms

Now, let's combine like terms in the expression.

25𝑥² + (−55𝑥) + 55𝑥 + (−121)

The terms −55𝑥 and 55𝑥 are like terms, so they can be combined.

25𝑥² + (−55𝑥 + 55𝑥) + (−121)

The expression −55𝑥 + 55𝑥 simplifies to 0, so the expression becomes:

25𝑥² + 0 + (−121)

The expression simplifies to:

25𝑥² − 121

Conclusion

The expanded form of the given algebraic expression (5𝑥 + 11)(5𝑥 − 11) is 25𝑥² − 121. We used the distributive property to expand the expression and then simplified each term. The final expression is a quadratic expression in the variable x.

Key Takeaways

  • The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term in one binomial with each term in another binomial.
  • When expanding expressions, we need to simplify each term and combine like terms to get the final expression.
  • The expanded form of the given expression is 25𝑥² − 121.

Common Identities Used

  • Distributive property: a(b + c) = ab + ac
  • Combining like terms: a + a = 2a, a + b + a + b = 2a + 2b

Real-World Applications

  • The distributive property is used in various real-world applications, such as finance, engineering, and computer science.
  • The concept of expanding expressions is used in solving systems of equations, graphing functions, and solving optimization problems.

Practice Problems

  • Expand the expression (3𝑥 + 2)(3𝑥 − 2)
  • Expand the expression (2𝑥 + 1)(2𝑥 − 1)
  • Expand the expression (4𝑥 + 3)(4𝑥 − 3)

Solutions

  • (3𝑥 + 2)(3𝑥 − 2) = 9𝑥² − 4
  • (2𝑥 + 1)(2𝑥 − 1) = 4𝑥² − 1
  • (4𝑥 + 3)(4𝑥 − 3) = 16𝑥² − 9
    Q&A: Expanding Algebraic Expressions

In this article, we will answer some frequently asked questions about expanding algebraic expressions. Whether you are a student, teacher, or simply someone who wants to learn more about algebra, this article is for you.

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term in one binomial with each term in another binomial. It states that for any real numbers a, b, and c, a(b + c) = ab + ac.

Q: How do I apply the distributive property?

A: To apply the distributive property, you need to multiply each term in the first binomial with each term in the second binomial. For example, to expand the expression (5𝑥 + 11)(5𝑥 − 11), you would multiply 5𝑥 with 5𝑥, 5𝑥 with −11, 11 with 5𝑥, and 11 with −11.

Q: What is the difference between expanding and simplifying an expression?

A: Expanding an expression involves multiplying each term in one binomial with each term in another binomial, while simplifying an expression involves combining like terms and eliminating any unnecessary terms.

Q: How do I simplify an expression?

A: To simplify an expression, you need to combine like terms and eliminate any unnecessary terms. For example, if you have the expression 2𝑥 + 3𝑥, you can combine the like terms to get 5𝑥.

Q: What are some common identities used in algebra?

A: Some common identities used in algebra include:

  • Distributive property: a(b + c) = ab + ac
  • Combining like terms: a + a = 2a, a + b + a + b = 2a + 2b
  • FOIL method: (a + b)(c + d) = ac + ad + bc + bd

Q: How do I use the FOIL method?

A: The FOIL method is a technique used to expand expressions of the form (a + b)(c + d). To use the FOIL method, you need to multiply the first terms in each binomial, then multiply the outer terms, then multiply the inner terms, and finally multiply the last terms.

Q: What are some real-world applications of expanding algebraic expressions?

A: Expanding algebraic expressions has many real-world applications, including:

  • Finance: Expanding expressions is used to calculate interest rates, investment returns, and other financial metrics.
  • Engineering: Expanding expressions is used to design and optimize systems, such as bridges, buildings, and electronic circuits.
  • Computer science: Expanding expressions is used to develop algorithms and data structures, such as sorting and searching algorithms.

Q: How can I practice expanding algebraic expressions?

A: There are many ways to practice expanding algebraic expressions, including:

  • Working through practice problems, such as those found in algebra textbooks or online resources.
  • Using online tools, such as algebra calculators or graphing software.
  • Creating your own practice problems and solving them.

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 distribute the terms correctly.
  • Not combining like terms.
  • Not eliminating unnecessary terms.

Conclusion

Expanding algebraic expressions is a fundamental concept in algebra that has many real-world applications. By understanding the distributive property and how to apply it, you can expand expressions and simplify them to get the final result. Remember to practice regularly and avoid common mistakes to become proficient in expanding algebraic expressions.