Simplifying The Expression 2 - 8(x + 1)^2 A Step-by-Step Guide
In this article, we will delve into the process of simplifying the algebraic expression 2 - 8(x + 1)^2. This expression involves a combination of arithmetic operations, including subtraction, multiplication, and exponentiation, along with the presence of a variable x. Simplifying such expressions is a fundamental skill in algebra, as it allows us to rewrite the expression in a more compact and manageable form, making it easier to analyze, evaluate, or use in further calculations. The ability to simplify algebraic expressions is crucial in various mathematical contexts, including solving equations, graphing functions, and modeling real-world phenomena. By mastering the techniques involved in simplifying expressions like this one, you'll be better equipped to tackle more complex algebraic problems and gain a deeper understanding of mathematical concepts. So, let's embark on this journey of simplification, breaking down the steps involved and revealing the underlying principles that govern the manipulation of algebraic expressions. The key here is to meticulously follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), and to apply algebraic identities and properties judiciously. By doing so, we can systematically transform the expression into its simplest form, uncovering its inherent structure and making it more accessible for further mathematical exploration.
Understanding the Order of Operations (PEMDAS/BODMAS)
Before we dive into simplifying the expression 2 - 8(x + 1)^2, it's crucial to understand the order of operations, a fundamental principle in mathematics that dictates the sequence in which operations should be performed. This order is often remembered by the acronym PEMDAS in the United States, which stands for: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). In some other regions, the acronym BODMAS is used, which stands for Brackets, Orders (powers and square roots, etc.), Division and Multiplication, Addition and Subtraction. Both PEMDAS and BODMAS represent the same order of operations. Ignoring this order can lead to incorrect results, so it's essential to adhere to it strictly. In our expression, we have parentheses, an exponent, multiplication, and subtraction. According to PEMDAS/BODMAS, we must first address the parentheses, then the exponent, followed by multiplication, and finally, subtraction. This systematic approach ensures that we simplify the expression correctly, step by step. The order of operations is not just a set of arbitrary rules; it's a logical framework that ensures mathematical consistency and allows us to communicate mathematical ideas unambiguously. By following PEMDAS/BODMAS, we can avoid confusion and arrive at the correct answer every time. This principle is not limited to simplifying algebraic expressions; it applies to all mathematical calculations, from basic arithmetic to advanced calculus. Therefore, mastering the order of operations is a cornerstone of mathematical proficiency, enabling us to navigate the world of numbers and symbols with confidence and accuracy. So, as we proceed with simplifying our expression, let's keep PEMDAS/BODMAS firmly in mind, guiding our steps and ensuring that we follow the correct sequence of operations.
Step-by-Step Simplification
Now, let's embark on the journey of simplifying the expression 2 - 8(x + 1)^2 step-by-step, meticulously following the order of operations. Our primary goal is to transform this expression into its simplest form, revealing its underlying structure and making it easier to understand and use. We'll break down the process into manageable steps, explaining the reasoning behind each operation and highlighting the key algebraic principles involved. This step-by-step approach will not only help us arrive at the correct answer but also deepen our understanding of the simplification process itself. Each step builds upon the previous one, creating a logical flow that leads us to the final simplified form. By carefully examining each step, we can gain insights into how algebraic expressions behave and how we can manipulate them effectively. This understanding is crucial for tackling more complex algebraic problems and for applying algebraic concepts in various mathematical contexts. So, let's begin our step-by-step simplification, armed with the knowledge of the order of operations and a commitment to precision and accuracy. Remember, the key to successful simplification is to be methodical and to pay close attention to the details of each operation. With patience and practice, you'll become proficient in simplifying algebraic expressions of all kinds.
Step 1: Expanding the Square
According to PEMDAS/BODMAS, we first need to address the parentheses. Inside the parentheses, we have the expression (x + 1), which is being squared. This means we need to expand (x + 1)^2. Expanding a squared binomial like this involves multiplying the binomial by itself: (x + 1)(x + 1). We can use the FOIL method (First, Outer, Inner, Last) or the distributive property to perform this multiplication. Applying the FOIL method, we multiply the First terms (x * x = x^2), the Outer terms (x * 1 = x), the Inner terms (1 * x = x), and the Last terms (1 * 1 = 1). Adding these together, we get x^2 + x + x + 1. Combining the like terms (the two x terms), we arrive at the expanded form: x^2 + 2x + 1. This expanded form is equivalent to (x + 1)^2 and is crucial for the next steps in simplifying the expression. Expanding the square is a fundamental algebraic operation that arises frequently in various mathematical contexts. Mastering this technique is essential for simplifying expressions, solving equations, and working with quadratic functions. The expanded form reveals the individual terms that make up the squared binomial, allowing us to manipulate them further and combine them with other terms in the expression. So, expanding the square is a crucial first step in our simplification journey, setting the stage for the subsequent operations.
Step 2: Distributing the -8
Now that we've expanded the square, our expression looks like this: 2 - 8(x^2 + 2x + 1). The next step, according to PEMDAS/BODMAS, is to address the multiplication. We have -8 multiplied by the trinomial (x^2 + 2x + 1). To perform this multiplication, we need to distribute the -8 to each term inside the parentheses. This means we multiply -8 by x^2, -8 by 2x, and -8 by 1. When we multiply -8 by x^2, we get -8x^2. When we multiply -8 by 2x, we get -16x. And when we multiply -8 by 1, we get -8. So, distributing the -8 gives us the expression -8x^2 - 16x - 8. This distribution step is crucial because it removes the parentheses and allows us to combine like terms in the next step. The distributive property is a fundamental principle in algebra that allows us to multiply a single term by a group of terms inside parentheses. It's a versatile tool that is used extensively in simplifying expressions, solving equations, and working with polynomials. Mastering the distributive property is essential for algebraic proficiency, enabling us to manipulate expressions effectively and accurately. So, distributing the -8 is a key step in our simplification process, transforming the expression and bringing us closer to its simplest form.
Step 3: Combining Like Terms
After distributing the -8, our expression now looks like this: 2 - 8x^2 - 16x - 8. The final step in simplifying the expression is to combine like terms. Like terms are terms that have the same variable raised to the same power. In our expression, we have two constant terms: 2 and -8. These are like terms because they don't have any variables attached to them. To combine them, we simply add them together: 2 + (-8) = -6. The other terms, -8x^2 and -16x, don't have any like terms in the expression, so they remain as they are. Therefore, combining the like terms gives us the simplified expression: -8x^2 - 16x - 6. This is the simplest form of the original expression, 2 - 8(x + 1)^2. Combining like terms is a fundamental algebraic operation that helps us to simplify expressions and make them more manageable. It involves identifying terms that have the same variable and exponent and then adding or subtracting their coefficients. This process reduces the number of terms in the expression and makes it easier to analyze, evaluate, or use in further calculations. Mastering the technique of combining like terms is essential for algebraic proficiency, enabling us to simplify expressions effectively and efficiently. So, combining like terms is the final step in our simplification journey, bringing us to the ultimate simplified form of the expression.
Final Simplified Expression
After meticulously following the order of operations and applying the necessary algebraic principles, we have successfully simplified the expression 2 - 8(x + 1)^2. Through a series of steps, including expanding the square, distributing, and combining like terms, we have arrived at the final simplified form: -8x^2 - 16x - 6. This expression is equivalent to the original expression but is written in a more compact and manageable form. It reveals the quadratic nature of the expression and makes it easier to analyze its properties, such as its roots, vertex, and axis of symmetry. The simplified expression also allows us to evaluate the expression for different values of x more efficiently. By substituting a value for x into the simplified expression, we can quickly calculate the corresponding value of the expression without having to perform the more complex operations involved in the original expression. Furthermore, the simplified expression is in a standard form that is commonly used in algebra and calculus. This makes it easier to compare the expression with other expressions and to apply various mathematical techniques to it. So, the final simplified expression, -8x^2 - 16x - 6, represents the culmination of our simplification journey, providing us with a more accessible and informative representation of the original expression. This process highlights the power of algebraic manipulation in transforming expressions into their simplest forms, making them easier to understand and use.
Conclusion
In conclusion, we have successfully simplified the expression 2 - 8(x + 1)^2 to its equivalent form, -8x^2 - 16x - 6. This process involved a careful application of the order of operations (PEMDAS/BODMAS), the expansion of the squared binomial, the distributive property, and the combination of like terms. By breaking down the simplification into manageable steps, we were able to systematically transform the expression into its simplest form, revealing its underlying structure and making it easier to analyze and use. The simplified expression, -8x^2 - 16x - 6, is a quadratic expression in standard form, which provides valuable insights into its properties and behavior. It allows us to easily identify the coefficients of the quadratic, linear, and constant terms, which are crucial for various mathematical operations, such as solving quadratic equations and graphing quadratic functions. Furthermore, the simplified expression is more efficient to evaluate for different values of x, making it a valuable tool in various mathematical and scientific applications. The ability to simplify algebraic expressions is a fundamental skill in mathematics, and this exercise demonstrates the importance of mastering the underlying principles and techniques involved. By understanding the order of operations, the distributive property, and the concept of like terms, we can confidently tackle more complex algebraic problems and gain a deeper appreciation for the power and elegance of mathematics. So, the journey of simplifying this expression has not only provided us with a concrete answer but has also reinforced our understanding of key algebraic concepts and skills, empowering us to approach future mathematical challenges with greater confidence and competence.