Select The Expression That Is Equivalent To $2(x-3)^2$.A. $2(x^2 - 6x + 9)$ B. $ 2 ( X 2 + 9 ) 2(x^2 + 9) 2 ( X 2 + 9 ) [/tex] C. $2x^2 - 12x + 18$ D. $4x^2 - 36$ E. $ 2 X 2 − 36 X + 9 2x^2 - 36x + 9 2 X 2 − 36 X + 9 [/tex]

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill to master. In this article, we will focus on simplifying a specific type of algebraic expression, namely the expression $2(x-3)^2$. We will explore the different options provided and determine which one is equivalent to the given expression.

Understanding the Expression

The given expression is $2(x-3)^2$. To simplify this expression, we need to understand the concept of expanding a squared binomial. When we expand a squared binomial, we use the formula $(a-b)^2 = a^2 - 2ab + b^2$.

Expanding the Expression

Using the formula, we can expand the expression $2(x-3)^2$ as follows:

2(x−3)2=2(x2−2⋅x⋅3+32)2(x-3)^2 = 2(x^2 - 2 \cdot x \cdot 3 + 3^2)

Simplifying the Expression

Now, we can simplify the expression by multiplying the constants and combining like terms:

2(x2−6x+9)=2x2−12x+182(x^2 - 6x + 9) = 2x^2 - 12x + 18

Analyzing the Options

Now that we have simplified the expression, let's analyze the options provided:

  • A. $2(x^2 - 6x + 9)$
  • B. $2(x^2 + 9)$
  • C. $2x^2 - 12x + 18$
  • D. $4x^2 - 36$
  • E. $2x^2 - 36x + 9$

Comparing the Options

Let's compare the simplified expression with the options provided:

  • Option A: $2(x^2 - 6x + 9)$ is equivalent to the simplified expression.
  • Option B: $2(x^2 + 9)$ is not equivalent to the simplified expression.
  • Option C: $2x^2 - 12x + 18$ is equivalent to the simplified expression.
  • Option D: $4x^2 - 36$ is not equivalent to the simplified expression.
  • Option E: $2x^2 - 36x + 9$ is not equivalent to the simplified expression.

Conclusion

Based on our analysis, we can conclude that the expression equivalent to $2(x-3)^2$ is:

  • A. $2(x^2 - 6x + 9)$
  • C. $2x^2 - 12x + 18$

Both options A and C are equivalent to the simplified expression. However, option C is a more simplified form of the expression.

Tips and Tricks

When simplifying algebraic expressions, it's essential to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

By following these steps, you can simplify complex algebraic expressions with ease.

Practice Problems

To practice simplifying algebraic expressions, try the following problems:

  1. Simplify the expression $3(2x-1)^2$.
  2. Simplify the expression $4(x+2)^2$.
  3. Simplify the expression $2(x-4)^2$.

Conclusion

Simplifying algebraic expressions is an essential skill in mathematics. By understanding the concept of expanding a squared binomial and following the order of operations, you can simplify complex expressions with ease. In this article, we explored the expression $2(x-3)^2$ and determined that the equivalent expression is $2(x^2 - 6x + 9)$ or $2x^2 - 12x + 18$. By practicing these skills, you can become proficient in simplifying algebraic expressions and tackle complex problems with confidence.

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Q: What is the order of operations in simplifying algebraic expressions?

A: The order of operations in simplifying algebraic expressions is:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I expand a squared binomial?

A: To expand a squared binomial, use the formula $(a-b)^2 = a^2 - 2ab + b^2$. For example, to expand $(x-3)^2$, you would use the formula as follows:

(x−3)2=x2−2⋅x⋅3+32(x-3)^2 = x^2 - 2 \cdot x \cdot 3 + 3^2

Q: What is the difference between a simplified expression and an equivalent expression?

A: A simplified expression is an expression that has been reduced to its simplest form, often by combining like terms or canceling out common factors. An equivalent expression, on the other hand, is an expression that has the same value as the original expression, but may not be in its simplest form.

Q: How do I determine if two expressions are equivalent?

A: To determine if two expressions are equivalent, you can use the following steps:

  1. Simplify both expressions to their simplest form.
  2. Compare the simplified expressions to see if they are the same.
  3. If the simplified expressions are the same, then the original expressions are equivalent.

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

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

  • Forgetting to follow the order of operations.
  • Not combining like terms.
  • Not canceling out common factors.
  • Not checking for equivalent expressions.

Q: How can I practice simplifying algebraic expressions?

A: You can practice simplifying algebraic expressions by:

  • Working through practice problems.
  • Using online resources and tools.
  • Asking a teacher or tutor for help.
  • Joining a study group or math club.

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

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

  • Solving equations and inequalities.
  • Graphing functions.
  • Modeling real-world situations.
  • Optimizing systems.

Q: How can I apply simplifying algebraic expressions to my everyday life?

A: You can apply simplifying algebraic expressions to your everyday life by:

  • Using algebra to solve problems in your personal and professional life.
  • Understanding and applying mathematical concepts to real-world situations.
  • Developing critical thinking and problem-solving skills.
  • Improving your communication and collaboration skills.

Conclusion

Simplifying algebraic expressions is an essential skill in mathematics, and it has many real-world applications. By understanding the order of operations, expanding squared binomials, and avoiding common mistakes, you can simplify complex expressions with ease. Practice problems, online resources, and real-world applications can help you develop your skills and apply them to your everyday life.