Equivalent Expressions Exploring 2x² - 2x + 7

#mainkeyword Which expression is equivalent to $2x^2 - 2x + 7$? This is a fundamental question in algebra that tests our understanding of combining like terms and simplifying polynomial expressions. In this article, we will delve into the process of identifying equivalent expressions, focusing specifically on the given expression $2x^2 - 2x + 7$. We will meticulously analyze each option, breaking down the steps involved in simplifying them and comparing the results with the target expression. This comprehensive exploration will not only provide the solution but also enhance your understanding of algebraic manipulations and equivalent forms. Grasping these concepts is crucial for success in higher-level mathematics and various real-world applications where algebraic expressions are used to model and solve problems.

Understanding Equivalent Expressions

Before diving into the options, let's first clarify what it means for two expressions to be equivalent. Equivalent expressions are expressions that, while possibly looking different, represent the same value for all possible values of the variable. In simpler terms, if you substitute any number for 'x' in two equivalent expressions, you will get the same result. To determine if expressions are equivalent, we typically simplify them by combining like terms. Like terms are terms that have the same variable raised to the same power (e.g., $2x^2$ and $-5x^2$ are like terms, while $2x^2$ and $2x$ are not). The process of combining like terms involves adding or subtracting the coefficients (the numerical part) of the terms while keeping the variable and its exponent the same. For example, $3x + 5x$ simplifies to $8x$. Understanding this basic principle is the cornerstone of identifying equivalent expressions. We will apply this principle repeatedly as we analyze each option and compare it to the target expression.

Analyzing Option A: $(4x + 12) + (2x^2 - 6x + 5)$

Let's begin by examining option A: $(4x + 12) + (2x^2 - 6x + 5)$. To determine if this expression is equivalent to $2x^2 - 2x + 7$, we need to simplify it by combining like terms. The first step is to remove the parentheses. Since we are adding the two expressions, we can simply rewrite the expression without parentheses: $4x + 12 + 2x^2 - 6x + 5$. Now, we identify the like terms. We have terms with $x^2$, terms with $x$, and constant terms (numbers without variables). The like terms are:

• $2x^2$ (there's only one term with $x^2$)
• $4x$ and $-6x$ (terms with $x$)
• $12$ and $5$ (constant terms)

Next, we combine these like terms. The $2x^2$ term remains as is. For the $x$ terms, we have $4x - 6x = -2x$. For the constant terms, we have $12 + 5 = 17$. Therefore, the simplified expression is $2x^2 - 2x + 17$. Now, we compare this simplified expression with the target expression, $2x^2 - 2x + 7$. We see that the $x^2$ and $x$ terms match, but the constant terms are different (17 versus 7). Thus, option A is not equivalent to the target expression.

Analyzing Option B: $(x^2 - 5x + 13) + (x^2 + 3x - 6)$

Now, let's consider option B: $(x^2 - 5x + 13) + (x^2 + 3x - 6)$. Similar to option A, we need to simplify this expression by combining like terms. We start by removing the parentheses: $x^2 - 5x + 13 + x^2 + 3x - 6$. Identifying the like terms, we have:

• $x^2$ and $x^2$ (terms with $x^2$)
• $-5x$ and $3x$ (terms with $x$)
• $13$ and $-6$ (constant terms)

Combining the like terms, we get: For the $x^2$ terms, $x^2 + x^2 = 2x^2$. For the $x$ terms, $-5x + 3x = -2x$. For the constant terms, $13 - 6 = 7$. So, the simplified expression is $2x^2 - 2x + 7$. Comparing this simplified expression with the target expression, $2x^2 - 2x + 7$, we see that they are identical. Therefore, option B is equivalent to the target expression. We have found our answer, but for the sake of completeness, let's analyze the remaining options as well.

Analyzing Option C: $(4x^2 - 6x + 11) + (2x^2 - 4x + 4)$

Next, we analyze option C: $(4x^2 - 6x + 11) + (2x^2 - 4x + 4)$. Removing the parentheses, we get: $4x^2 - 6x + 11 + 2x^2 - 4x + 4$. Identifying and combining like terms:

• $4x^2 + 2x^2 = 6x^2$ (terms with $x^2$)
• $-6x - 4x = -10x$ (terms with $x$)
• $11 + 4 = 15$ (constant terms)

The simplified expression is $6x^2 - 10x + 15$. Comparing this with the target expression, $2x^2 - 2x + 7$, we see that none of the terms match. Thus, option C is not equivalent to the target expression.

Analyzing Option D: $(5x^2 - 8x + 120) + (-3x^2 + 10x - 13)$

Finally, let's analyze option D: $(5x^2 - 8x + 120) + (-3x^2 + 10x - 13)$. Removing the parentheses, we have: $5x^2 - 8x + 120 - 3x^2 + 10x - 13$. Identifying and combining like terms:

• $5x^2 - 3x^2 = 2x^2$ (terms with $x^2$)
• $-8x + 10x = 2x$ (terms with $x$)
• $120 - 13 = 107$ (constant terms)

The simplified expression is $2x^2 + 2x + 107$. Comparing this with the target expression, $2x^2 - 2x + 7$, we see that the $x^2$ term matches, but the $x$ and constant terms are different. Therefore, option D is not equivalent to the target expression.

Conclusion

In conclusion, after meticulously analyzing each option by simplifying the expressions and comparing them to the target expression $2x^2 - 2x + 7$, we found that option B, $(x^2 - 5x + 13) + (x^2 + 3x - 6)$, is the only equivalent expression. This exercise highlights the importance of understanding how to combine like terms and simplify algebraic expressions. The ability to identify equivalent expressions is a fundamental skill in algebra and is crucial for solving more complex mathematical problems. By mastering these techniques, you can confidently tackle a wide range of algebraic challenges and build a strong foundation for future mathematical endeavors.

This exploration provides a comprehensive understanding of how to determine equivalent expressions, emphasizing the critical role of simplification and comparison. The detailed analysis of each option not only reveals the correct answer but also reinforces the underlying principles of algebraic manipulation. This knowledge will empower you to approach similar problems with greater confidence and accuracy, ensuring success in your mathematical journey.