Equivalent Expression Of 2x² - 2x + 7

In the realm of algebra, quadratic expressions form the bedrock of numerous mathematical concepts and applications. Grasping how to manipulate and recognize equivalent forms of these expressions is a crucial skill for students and professionals alike. This article delves into the process of identifying expressions equivalent to a given quadratic, specifically focusing on the expression 2x² - 2x + 7. We will dissect the problem, explore various approaches, and ultimately arrive at the solution, providing a comprehensive understanding of the underlying principles.

Understanding Quadratic Expressions

Before we embark on our quest to find the equivalent expression, it's essential to solidify our understanding of what a quadratic expression truly is. A quadratic expression is a polynomial expression of degree two. This means that the highest power of the variable (usually x) is 2. The general form of a quadratic expression is ax² + bx + c, where a, b, and c are constants, and a is not equal to zero. The coefficient a determines the parabola's direction, the coefficient b influences the parabola's position, and the constant c indicates the y-intercept. The expression 2x² - 2x + 7 perfectly fits this form, with a = 2, b = -2, and c = 7.

Identifying Equivalent Expressions: The Core Principle

The crux of the matter lies in recognizing that equivalent expressions, while appearing different on the surface, yield the same value for all possible values of the variable x. In simpler terms, if we were to substitute any number for x in the original expression and its equivalent counterpart, the results would be identical. This principle guides our approach to solving the problem. We aim to manipulate the given options, combining like terms, to see which one collapses into the original expression, 2x² - 2x + 7. This process of simplification is fundamental to algebra, allowing us to work with expressions in their most manageable form. The ability to recognize and create equivalent expressions is a cornerstone of algebraic manipulation, providing the flexibility needed to solve equations, graph functions, and model real-world phenomena. Therefore, mastering this skill is not just about answering test questions but about building a solid foundation for future mathematical endeavors.

The Problem: Which Expression Matches 2x² - 2x + 7?

Our mission is clear: identify the expression that, upon simplification, mirrors the given quadratic, 2x² - 2x + 7. We are presented with four options, each a sum of two polynomial expressions. Our strategy involves systematically simplifying each option by combining like terms and then comparing the result with the target expression. This methodical approach ensures we don't overlook any subtle nuances in the algebraic manipulations. The options presented are designed to test not only our understanding of combining like terms but also our attention to detail in managing signs and coefficients. Therefore, a careful, step-by-step approach is paramount to success. Let's delve into each option, applying the principles of algebraic simplification to unearth the equivalent expression.

A. (4x + 12) + (2x² - 6x + 5)

The first contender is the expression (4x + 12) + (2x² - 6x + 5). To assess its equivalence to 2x² - 2x + 7, we embark on the process of simplification. Our initial step involves removing the parentheses, which is straightforward in this case since we are dealing with addition. This yields 4x + 12 + 2x² - 6x + 5. Now, the task is to identify and combine like terms. Like terms are those that share the same variable raised to the same power. In our expression, we have x² terms, x terms, and constant terms. The lone x² term is 2x². For the x terms, we have 4x and -6x, which combine to give -2x. The constant terms are 12 and 5, summing up to 17. Thus, the simplified expression becomes 2x² - 2x + 17. Comparing this to our target expression, 2x² - 2x + 7, we observe a discrepancy in the constant term. While the x² and x terms match perfectly, the constant term is 17 in the simplified expression, not 7. This crucial difference leads us to conclude that option A is not equivalent to the given quadratic.

B. (x² - 5x + 13) + (x² + 3x - 6)

Moving on to option B, we have the expression (x² - 5x + 13) + (x² + 3x - 6). As before, our initial maneuver involves removing the parentheses, resulting in x² - 5x + 13 + x² + 3x - 6. The next step is the meticulous combination of like terms. We identify two x² terms: x² and x², which combine to give 2x². For the x terms, we have -5x and 3x, summing up to -2x. The constant terms are 13 and -6, which combine to yield 7. Therefore, the simplified form of option B is 2x² - 2x + 7. A direct comparison with our target expression, 2x² - 2x + 7, reveals a perfect match. All the terms, including the x² term, the x term, and the constant term, align precisely. This congruence confirms that option B is indeed equivalent to the given quadratic expression. We have successfully identified one equivalent expression, but to ensure our understanding and demonstrate thoroughness, let's examine the remaining options as well.

C. (4x² - 6x + 11) + (2x² - 4x + 4)

Now, let's dissect option C: (4x² - 6x + 11) + (2x² - 4x + 4). We commence by eliminating the parentheses, resulting in 4x² - 6x + 11 + 2x² - 4x + 4. The subsequent step involves the systematic combination of like terms. We identify two x² terms: 4x² and 2x², which coalesce to form 6x². For the x terms, we have -6x and -4x, yielding a sum of -10x. The constant terms are 11 and 4, culminating in 15. Consequently, the simplified expression for option C is 6x² - 10x + 15. Juxtaposing this with our target expression, 2x² - 2x + 7, we observe significant disparities. The coefficients of the x² and x terms, as well as the constant term, are all distinct. This stark divergence unequivocally indicates that option C is not equivalent to the given quadratic. The x² term alone, 6x², clearly deviates from the 2x² in the original expression, precluding any possibility of equivalence. This reinforces the importance of meticulously comparing all terms when determining equivalence.

D. (5x² - 8x + 120) + (-3x² + 10x - 13)

Finally, we turn our attention to option D: (5x² - 8x + 120) + (-3x² + 10x - 13). As before, we begin by removing the parentheses, obtaining 5x² - 8x + 120 - 3x² + 10x - 13. The next step, as always, is to diligently combine like terms. We identify two x² terms: 5x² and -3x², which combine to give 2x². For the x terms, we have -8x and 10x, summing up to 2x. The constant terms are 120 and -13, resulting in 107. Thus, the simplified form of option D is 2x² + 2x + 107. Comparing this to our target expression, 2x² - 2x + 7, we observe disparities in both the x term and the constant term. While the x² term matches, the x term is +2x in option D, whereas it is -2x in the original expression. Additionally, the constant term is 107 in option D, a far cry from the 7 in the target expression. These discrepancies definitively demonstrate that option D is not equivalent to the given quadratic.

Conclusion: The Equivalent Expression

Through our systematic analysis of each option, we have definitively identified the expression equivalent to 2x² - 2x + 7. By meticulously simplifying each option and comparing the result with the target quadratic, we found that option B, (x² - 5x + 13) + (x² + 3x - 6), perfectly matches the original expression upon simplification. This exercise underscores the importance of understanding the fundamental principles of algebraic manipulation, particularly the combination of like terms. The ability to recognize and create equivalent expressions is a cornerstone of mathematical proficiency, enabling us to solve equations, graph functions, and tackle a wide array of mathematical challenges. Remember, the key is to approach these problems methodically, paying close attention to signs, coefficients, and the order of operations. With practice and a solid grasp of the underlying concepts, you can confidently navigate the world of quadratic expressions and beyond.