Are you grappling with algebraic expressions and trying to find equivalent forms? You're not alone! Many students and math enthusiasts often encounter the challenge of simplifying and identifying equivalent expressions. In this comprehensive guide, we will dissect the expression 2x² - 2x + 7 and meticulously examine the provided options to pinpoint the one that matches. We will walk through each option step-by-step, ensuring you grasp the underlying principles of combining like terms and simplifying polynomial expressions. By the end of this article, you'll not only know the correct answer but also have a deeper understanding of how to tackle similar problems with confidence. Let's dive in and unravel the world of algebraic equivalency!
Understanding Equivalent Expressions
Before we embark on our quest to find the equivalent expression for 2x² - 2x + 7, it's crucial to establish a firm understanding of what equivalent expressions truly mean. In mathematics, equivalent expressions are expressions that, despite their potentially different appearances, yield the same value for every possible value of the variable. Think of them as different outfits that ultimately reveal the same person underneath. For instance, the expressions x + x and 2x are equivalent because no matter what value you substitute for x, both expressions will always produce the same result. This fundamental concept of equivalency is the cornerstone of algebraic manipulation and simplification.
So, how do we determine if two expressions are equivalent? The key lies in the process of simplification. We meticulously combine like terms, which are terms that share the same variable raised to the same power. For example, in the expression 3x² + 2x - x² + 5x, the terms 3x² and -x² are like terms because they both involve x², while 2x and 5x are like terms because they both involve x. The constant terms, which are the numbers without any variables, are also like terms. By combining these like terms, we can often transform a complex-looking expression into a simpler, equivalent form. In the aforementioned example, combining like terms would yield 2x² + 7x, a simplified expression that is equivalent to the original.
The process of identifying equivalent expressions often involves a combination of algebraic techniques, including the distributive property, the commutative property, and the associative property. The distributive property allows us to multiply a term across a sum or difference, such as a(b + c) = ab + ac. The commutative property states that the order of addition or multiplication doesn't affect the result, meaning a + b = b + a and a * b = b * a. The associative property states that the grouping of terms in addition or multiplication doesn't affect the result, meaning (a + b) + c = a + (b + c) and (a * b) * c = a * (b * c). Mastering these properties is essential for confidently navigating the world of algebraic expressions and determining equivalency.
Analyzing the Target Expression: 2x² - 2x + 7
Our primary objective is to identify which of the given options is equivalent to the expression 2x² - 2x + 7. Before we delve into the options, let's take a closer look at our target expression. This expression is a quadratic trinomial, which means it's a polynomial with three terms, and the highest power of the variable x is 2. The expression consists of three distinct terms: a quadratic term (2x²), a linear term (-2x), and a constant term (7). It's crucial to recognize these components as they will guide us in our comparison process.
The quadratic term, 2x², signifies that the variable x is squared and then multiplied by the coefficient 2. This term dictates the parabolic shape of the graph if we were to plot this expression as a function. The linear term, -2x, indicates that the variable x is multiplied by the coefficient -2. This term contributes to the slope and direction of the graph. The constant term, 7, is simply a numerical value without any variable attached. It represents the y-intercept of the graph, the point where the parabola intersects the vertical axis.
Now that we have a clear understanding of the components of our target expression, we can use this knowledge as a benchmark against which we will evaluate the given options. We are looking for an expression that, after simplification, yields precisely 2x² - 2x + 7. This means that the quadratic terms must add up to 2x², the linear terms must add up to -2x, and the constant terms must add up to 7. Any deviation from these values would disqualify an option as being equivalent. As we analyze each option, we will meticulously track the coefficients and constants, ensuring they align with our target expression.
Evaluating Option 1: (4x + 12) + (2x² - 6x + 5)
The first option we'll examine is (4x + 12) + (2x² - 6x + 5). To determine if this expression is equivalent to our target expression, 2x² - 2x + 7, we need to simplify it by combining like terms. The first step in this process is to remove the parentheses. Since we are adding the two expressions, the parentheses can be removed without altering the signs of the terms inside. This gives us: 4x + 12 + 2x² - 6x + 5.
Next, we identify and combine the like terms. We have a quadratic term (2x²), linear terms (4x and -6x), and constant terms (12 and 5). Combining the linear terms, 4x - 6x, we get -2x. Combining the constant terms, 12 + 5, we get 17. Now, let's rewrite the expression with the combined like terms, arranging the terms in descending order of their exponents: 2x² - 2x + 17.
Now, we compare the simplified expression, 2x² - 2x + 17, with our target expression, 2x² - 2x + 7. Notice that the quadratic term (2x²) and the linear term (-2x) match perfectly. However, the constant term in the simplified expression (17) differs from the constant term in our target expression (7). Since the constant terms do not match, we can definitively conclude that (4x + 12) + (2x² - 6x + 5) is not equivalent to 2x² - 2x + 7. This option can be eliminated from our list of potential solutions.
Evaluating Option 2: (x² - 5x + 13) + (x² + 3x - 6)
Our second contender for equivalency is the expression (x² - 5x + 13) + (x² + 3x - 6). Following the same procedure as before, our initial step is to simplify the expression by combining like terms. We begin by removing the parentheses. As with the previous option, we are adding the expressions, so the parentheses can be removed without changing the signs of the terms inside. This gives us: x² - 5x + 13 + x² + 3x - 6.
Now, we identify and combine the like terms. We have quadratic terms (x² and x²), linear terms (-5x and 3x), and constant terms (13 and -6). Combining the quadratic terms, x² + x², we get 2x². Combining the linear terms, -5x + 3x, we get -2x. Combining the constant terms, 13 - 6, we get 7. Now, we rewrite the expression with the combined like terms: 2x² - 2x + 7.
Now, the moment of truth! We compare the simplified expression, 2x² - 2x + 7, with our target expression, 2x² - 2x + 7. Lo and behold, the simplified expression perfectly matches our target expression. The quadratic terms, linear terms, and constant terms all align. This indicates that (x² - 5x + 13) + (x² + 3x - 6) is indeed equivalent to 2x² - 2x + 7. We have found a potential solution!
However, to be absolutely certain, it's prudent to evaluate the remaining options. Even though we have found a matching expression, there could potentially be another one that is also equivalent. Let's proceed with the analysis of the next option to ensure we have identified the sole equivalent expression.
Evaluating Option 3: (4x² - 6x + 11) + (-2x² - 4x + 4)
Now let's scrutinize the third option: (4x² - 6x + 11) + (-2x² - 4x + 4). As before, our initial step involves simplifying the expression by combining like terms. We begin by removing the parentheses. Since we are adding the expressions, we can remove the parentheses without altering any signs, resulting in: 4x² - 6x + 11 - 2x² - 4x + 4.
Next, we identify and combine the like terms. We have quadratic terms (4x² and -2x²), linear terms (-6x and -4x), and constant terms (11 and 4). Combining the quadratic terms, 4x² - 2x², we obtain 2x². Combining the linear terms, -6x - 4x, we get -10x. Combining the constant terms, 11 + 4, we get 15. Now, we rewrite the expression with the combined like terms: 2x² - 10x + 15.
We now compare this simplified expression, 2x² - 10x + 15, with our target expression, 2x² - 2x + 7. While the quadratic term (2x²) matches, the linear term (-10x) and the constant term (15) do not match the corresponding terms in our target expression (-2x and 7, respectively). This discrepancy definitively indicates that (4x² - 6x + 11) + (-2x² - 4x + 4) is not equivalent to 2x² - 2x + 7. We can confidently eliminate this option.
Evaluating Option 4: (5x² - 8x + 120) + (-3x² + 10x - 13)
Our final option to evaluate is (5x² - 8x + 120) + (-3x² + 10x - 13). Following our established procedure, we begin by simplifying the expression through combining like terms. We remove the parentheses, noting that we are adding the expressions, so the signs remain unchanged: 5x² - 8x + 120 - 3x² + 10x - 13.
Next, we identify and combine the like terms. We have quadratic terms (5x² and -3x²), linear terms (-8x and 10x), and constant terms (120 and -13). Combining the quadratic terms, 5x² - 3x², we get 2x². Combining the linear terms, -8x + 10x, we get 2x. Combining the constant terms, 120 - 13, we get 107. Now, we rewrite the expression with the combined like terms: 2x² + 2x + 107.
Finally, we compare this simplified expression, 2x² + 2x + 107, with our target expression, 2x² - 2x + 7. In this case, only the quadratic term (2x²) matches. The linear term (2x) and the constant term (107) are significantly different from the corresponding terms in our target expression (-2x and 7, respectively). This unequivocally confirms that (5x² - 8x + 120) + (-3x² + 10x - 13) is not equivalent to 2x² - 2x + 7. This option can be eliminated.
Conclusion: The Equivalent Expression
After a thorough and meticulous evaluation of all four options, we have arrived at a definitive conclusion. By systematically simplifying each expression and comparing it to our target expression, 2x² - 2x + 7, we were able to identify the sole equivalent expression.
Our analysis revealed that option 2, (x² - 5x + 13) + (x² + 3x - 6), is the only expression that, when simplified, perfectly matches our target expression. By combining like terms, we transformed (x² - 5x + 13) + (x² + 3x - 6) into 2x² - 2x + 7, thus confirming its equivalency.
Therefore, the final answer to the question, "Which expression is equivalent to 2x² - 2x + 7?" is:
(x² - 5x + 13) + (x² + 3x - 6)
This comprehensive guide has not only provided the solution but has also walked you through the process of identifying equivalent expressions. By understanding the principles of combining like terms and carefully comparing coefficients and constants, you can confidently tackle similar problems in the future. Remember, the key to success in algebra lies in a methodical approach and a solid grasp of fundamental concepts.