Equivalent Expressions For 2x^2 - 2x + 7 A Comprehensive Guide

In the realm of algebra, identifying equivalent expressions is a fundamental skill. These are expressions that, despite appearing different, yield the same value for any given variable. This article delves into the process of determining which expression is equivalent to the quadratic expression $2x^2 - 2x + 7$. We'll explore various algebraic manipulations and simplification techniques to arrive at the correct answer, ensuring a comprehensive understanding of the underlying concepts.

At the heart of this exploration lies the ability to combine like terms. Like terms are those that share the same variable raised to the same power. For instance, $3x^2$ and $-5x^2$ are like terms because they both contain the variable $x$ raised to the power of 2. Similarly, $7x$ and $-2x$ are like terms, while 8 and -3 are constant like terms. Combining like terms involves adding or subtracting their coefficients, the numerical part of the term, while keeping the variable part unchanged. This process is crucial for simplifying algebraic expressions and revealing their underlying equivalence.

The distributive property is another cornerstone of algebraic manipulation. It states that multiplying a sum or difference by a number is the same as multiplying each term inside the parentheses by that number individually. Mathematically, this is expressed as $a(b + c) = ab + ac$ and $a(b - c) = ab - ac$. The distributive property is essential for expanding expressions and removing parentheses, which often reveals hidden like terms that can then be combined. Mastering this property is vital for simplifying complex expressions and uncovering their equivalence to simpler forms.

To ascertain if two expressions are indeed equivalent, we employ the strategy of simplifying each expression to its simplest form and then comparing the results. If the simplified forms are identical, then the original expressions are equivalent. This method hinges on the principles of combining like terms and applying the distributive property, effectively reducing complex expressions to their most basic and comparable forms. By meticulously simplifying each expression, we can confidently determine whether they represent the same mathematical relationship.

Analyzing the Given Expressions

We are presented with the target expression $2x^2 - 2x + 7$ and a set of potential equivalents. Our mission is to meticulously analyze each of the provided options, simplify them using algebraic principles, and then compare the simplified results with our target expression. This process will involve a careful application of the distributive property, the identification and combination of like terms, and a steadfast commitment to maintaining the integrity of the mathematical operations.

Option 1: $(4x + 12) + (2x^2 - 6x + 5)$

To simplify this expression, our initial step is to remove the parentheses. Since we are adding the two expressions, the parentheses can be removed directly without altering any signs. This yields: $4x + 12 + 2x^2 - 6x + 5$. Next, we identify and combine like terms. The like terms are $4x$ and $-6x$, which combine to $-2x$, and the constants 12 and 5, which combine to 17. The $2x^2$ term remains unchanged as there are no other terms with $x^2$. Thus, the simplified expression is $2x^2 - 2x + 17$. Comparing this with our target expression, $2x^2 - 2x + 7$, we observe a discrepancy in the constant term. Therefore, this option is not equivalent.

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

As in the previous case, we begin by removing the parentheses, which gives us: $x^2 - 5x + 13 + x^2 + 3x - 6$. Now, we identify and combine the like terms. The $x^2$ terms combine to $2x^2$, the $x$ terms (-5x and 3x) combine to $-2x$, and the constant terms (13 and -6) combine to 7. The resulting simplified expression is $2x^2 - 2x + 7$. When compared with our target expression, $2x^2 - 2x + 7$, we find a perfect match. This indicates that this expression is indeed equivalent to the target expression.

Option 3: $(4x^2 - 6x + 11) + (2x^2 - 4x + 4)$

Removing the parentheses, we have: $4x^2 - 6x + 11 + 2x^2 - 4x + 4$. Combining like terms, the $x^2$ terms (4x^2 and 2x^2) combine to $6x^2$, the $x$ terms (-6x and -4x) combine to $-10x$, and the constant terms (11 and 4) combine to 15. The simplified expression is $6x^2 - 10x + 15$. This expression clearly differs from our target expression, $2x^2 - 2x + 7$, particularly in the coefficient of the $x^2$ term. Hence, this option is not equivalent.

Option 4: $(5x^2 - 8x + 3) - (3x^2 - 6x - 4)$

This option introduces a subtraction between the two expressions, necessitating careful attention to the distribution of the negative sign. Removing the parentheses, we must distribute the negative sign to each term in the second expression: $5x^2 - 8x + 3 - 3x^2 + 6x + 4$. Now, we combine like terms. The $x^2$ terms (5x^2 and -3x^2) combine to $2x^2$, the $x$ terms (-8x and 6x) combine to $-2x$, and the constant terms (3 and 4) combine to 7. The simplified expression is $2x^2 - 2x + 7$. This simplified expression perfectly matches our target expression, $2x^2 - 2x + 7$. Therefore, this expression is also equivalent to the target.

Conclusion: Identifying Equivalent Algebraic Expressions

Through meticulous simplification and comparison, we've successfully identified the expressions equivalent to $2x^2 - 2x + 7$. Options 2 and 4, after simplification, resulted in the exact same expression, demonstrating their equivalence. This exercise underscores the importance of mastering algebraic manipulation techniques such as combining like terms and applying the distributive property.

This exploration not only answers the specific question but also reinforces the broader principle of algebraic equivalence. Understanding how to identify and manipulate equivalent expressions is crucial for solving equations, simplifying complex formulas, and building a robust foundation in algebra. By consistently practicing these techniques, students can develop a deeper appreciation for the elegance and power of algebraic reasoning. The ability to transform expressions while preserving their underlying value is a cornerstone of mathematical proficiency, opening doors to more advanced concepts and problem-solving strategies.