Equivalent Expressions To (2x + 5)(3x - 2) A Step By Step Guide

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In the realm of algebra, simplifying and manipulating expressions is a fundamental skill. When faced with an expression like (2x + 5)(3x - 2), identifying its equivalent forms becomes crucial for solving equations, graphing functions, and understanding mathematical relationships. This article delves into the process of finding equivalent expressions, offering a step-by-step approach and exploring common techniques. We'll dissect the given expression, explore different methods of expansion, and analyze the provided options to pinpoint the correct equivalent form. Understanding these concepts is not just about finding the right answer; it's about building a solid foundation in algebraic manipulation, which is essential for more advanced mathematical concepts.

Deciphering the Expression (2x + 5)(3x - 2)

The expression (2x + 5)(3x - 2) represents the product of two binomials. To find an equivalent expression, we need to expand this product using the distributive property. This property, a cornerstone of algebra, dictates how multiplication interacts with addition and subtraction within parentheses. In essence, it allows us to break down complex expressions into simpler terms. The distributive property states that a(b + c) = ab + ac. Applying this to our expression, we'll multiply each term in the first binomial by each term in the second binomial. This process, often visualized using the acronym FOIL (First, Outer, Inner, Last), ensures that every term is accounted for in the expansion. By meticulously applying the distributive property, we can transform the original expression into a sum of individual terms, revealing its underlying structure and paving the way for simplification and further manipulation.

Applying the Distributive Property (FOIL Method)

The FOIL method is a mnemonic acronym that helps us remember the steps for multiplying two binomials: First, Outer, Inner, Last. Let's apply this method to expand (2x + 5)(3x - 2):

  • First: Multiply the first terms of each binomial: (2x)(3x) = 6x²
  • Outer: Multiply the outer terms of the binomials: (2x)(-2) = -4x
  • Inner: Multiply the inner terms of the binomials: (5)(3x) = 15x
  • Last: Multiply the last terms of each binomial: (5)(-2) = -10

Now, we add these products together: 6x² - 4x + 15x - 10. This resulting expression is a direct expansion of the original product, but it's not yet in its simplest form. To fully unveil the equivalent expression, we need to combine like terms. Like terms, in algebraic parlance, are terms that contain the same variable raised to the same power. Combining like terms is a crucial step in simplifying expressions, allowing us to consolidate terms and present the expression in a more concise and manageable form. By identifying and combining like terms, we move closer to the final equivalent expression.

Simplifying the Expanded Expression

After applying the FOIL method, we have the expression 6x² - 4x + 15x - 10. Notice that -4x and 15x are like terms because they both contain the variable 'x' raised to the power of 1. To simplify, we combine these terms by adding their coefficients: -4 + 15 = 11. This gives us the simplified expression 6x² + 11x - 10. This expression is equivalent to the original expression (2x + 5)(3x - 2), but it's written in a more standard and simplified form. The process of simplification is essential in algebra as it allows us to work with expressions more easily, solve equations, and understand the relationships between different mathematical quantities. The simplified form, 6x² + 11x - 10, reveals the quadratic nature of the expression and provides a clear representation of its terms.

Analyzing the Options

Now, let's analyze the given options to determine which one is equivalent to our simplified expression, 6x² + 11x - 10.

  • A. (2x)(3x) + (5)(-2): This option only multiplies the first terms and the last terms of the binomials. It omits the outer and inner products, which are crucial for a correct expansion. Therefore, this option is incorrect.
  • B. 2x(3x - 2) + 5(3x - 2): This option correctly applies the distributive property by multiplying each term of the first binomial by the entire second binomial. This is a valid approach to expanding the expression and is likely the correct answer.
  • C. 2(3x - 2)(3x - 2): This option is incorrect because it squares the second binomial and multiplies the result by 2. This does not follow the correct procedure for expanding the product of two different binomials.
  • D. (2x)(3x) + 5(3x - 2): This option multiplies the first terms correctly but only distributes the 5 to the second binomial. It misses the term that results from multiplying -2 by 2x, making it an incomplete expansion and therefore incorrect.

By carefully examining each option and comparing it to the correct expansion process and the simplified expression, we can confidently identify the equivalent form.

Identifying the Correct Equivalent Expression

Based on our analysis, Option B, 2x(3x - 2) + 5(3x - 2), is the correct equivalent expression. This option demonstrates the application of the distributive property, a fundamental principle in algebra. It accurately represents the expansion of (2x + 5)(3x - 2) by multiplying each term of the first binomial with the entire second binomial. This method ensures that all terms are accounted for, leading to a correct representation of the original expression. The other options, in contrast, either omit crucial terms or misapply the distributive property, resulting in incorrect expansions. Option B, therefore, stands as the accurate equivalent expression, showcasing a proper understanding of algebraic manipulation.

Conclusion: Mastering Equivalent Expressions

In conclusion, the expression 2x(3x - 2) + 5(3x - 2), Option B, is the same as (2x + 5)(3x - 2). This exercise highlights the importance of understanding and applying the distributive property, particularly the FOIL method, when expanding binomial expressions. Identifying equivalent expressions is a fundamental skill in algebra, essential for simplifying equations, solving problems, and building a strong foundation for more advanced mathematical concepts. By mastering these techniques, students can confidently navigate algebraic manipulations and unlock a deeper understanding of mathematical relationships. The ability to recognize and generate equivalent expressions is not just about finding the right answer; it's about developing a flexible and adaptable approach to problem-solving in mathematics.

By understanding these principles and practicing these techniques, you can confidently tackle similar problems and deepen your understanding of algebraic expressions.