Solution Set Of The Equation Y² - 49 = 0
In mathematics, solving equations is a fundamental skill. This article aims to provide a detailed explanation of how to find the solution set for the quadratic equation y² - 49 = 0. We will explore various methods to solve this equation, ensuring a comprehensive understanding for readers. Quadratic equations, characterized by the highest power of the variable being 2, appear frequently in various mathematical and real-world applications. Mastering the techniques to solve these equations is crucial for anyone studying algebra, calculus, or related fields. The equation y² - 49 = 0 is a classic example of a difference of squares, which simplifies the solving process considerably. We will delve into this method, as well as alternative approaches, to ensure a thorough understanding. This article is structured to guide you step-by-step through the solution process, offering clear explanations and examples along the way. Whether you are a student learning algebra for the first time or someone looking to refresh your skills, this guide will provide you with the tools and knowledge to confidently solve this type of equation. Understanding the underlying principles and techniques will empower you to tackle more complex mathematical problems in the future. So, let's embark on this mathematical journey together and unlock the secrets of solving quadratic equations!
Understanding the Equation
Before diving into the solution, it’s crucial to understand the equation y² - 49 = 0. This is a quadratic equation, a polynomial equation of degree 2. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and x is the variable. In our case, the equation can be seen as 1y² + 0y - 49 = 0, where a = 1, b = 0, and c = -49. Recognizing this structure is the first step in choosing the appropriate method for solving the equation. One key observation is that the equation is a difference of squares. The expression y² - 49 can be written as y² - 7². The difference of squares factorization is a powerful tool for simplifying and solving quadratic equations. The formula for the difference of squares is a² - b² = (a - b)(a + b). Applying this to our equation, we can rewrite y² - 49 as (y - 7)(y + 7). This factorization transforms the equation into a product of two binomials, making it easier to find the solutions. Understanding this transformation is vital, as it allows us to apply the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This property is the cornerstone of solving many algebraic equations, especially quadratic equations. By setting each factor equal to zero, we can isolate the variable and find the values that satisfy the original equation. In the following sections, we will explore this method in detail, along with other approaches to solving the equation y² - 49 = 0. The ability to recognize patterns and apply appropriate techniques is a hallmark of mathematical proficiency.
Method 1: Factoring (Difference of Squares)
The most straightforward method to solve y² - 49 = 0 is by factoring, specifically using the difference of squares formula. As mentioned earlier, the equation can be rewritten as y² - 7² = 0. Applying the difference of squares formula, a² - b² = (a - b)(a + b), we get (y - 7)(y + 7) = 0. This factorization is crucial because it breaks down the quadratic equation into a product of two linear factors. Each factor represents a possible solution to the equation. Now, we apply the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. Mathematically, if AB = 0, then either A = 0 or B = 0 (or both). In our case, the factors are (y - 7) and (y + 7). Setting each factor equal to zero gives us two separate linear equations:
- y - 7 = 0
- y + 7 = 0
Solving the first equation, y - 7 = 0, we add 7 to both sides to isolate y: y = 7. Solving the second equation, y + 7 = 0, we subtract 7 from both sides to isolate y: y = -7. Therefore, the solutions to the equation y² - 49 = 0 are y = 7 and y = -7. These values are the roots or zeros of the quadratic equation. Factoring is a powerful technique that simplifies the process of solving quadratic equations. It relies on recognizing patterns and applying algebraic identities to break down complex expressions into simpler forms. The difference of squares is a particularly useful pattern to recognize, as it appears frequently in various mathematical contexts. By mastering this method, you can efficiently solve a wide range of quadratic equations.
Method 2: Square Root Property
Another effective method for solving the equation y² - 49 = 0 is using the square root property. This method is particularly useful when the equation is in the form x² = k, where k is a constant. Our equation, y² - 49 = 0, can be easily transformed into this form. First, we add 49 to both sides of the equation:
y² = 49
Now, we have the equation in the form y² = k, where k = 49. The square root property states that if x² = k, then x = ±√k. This is because both the positive and negative square roots of k, when squared, will result in k. Applying this property to our equation, we take the square root of both sides:
√(y²) = ±√49
This simplifies to:
y = ±7
This gives us two solutions: y = 7 and y = -7. These are the same solutions we found using the factoring method, reinforcing the accuracy of both approaches. The square root property is a direct and efficient way to solve quadratic equations when the variable term is squared and isolated. It bypasses the need for factoring or using the quadratic formula, making it a preferred method for certain types of equations. However, it’s crucial to remember the ± sign when taking the square root, as both positive and negative roots satisfy the equation. The square root property highlights the fundamental relationship between squaring and taking square roots, which is a cornerstone of algebraic manipulation. By understanding and applying this property, you can quickly and accurately solve a variety of quadratic equations.
Solution Set
Having solved the equation y² - 49 = 0 using two different methods—factoring and the square root property—we have arrived at the same solutions: y = 7 and y = -7. The solution set is the collection of all values that satisfy the equation. In this case, the solution set consists of two elements, 7 and -7. We can represent the solution set using set notation as {-7, 7}. This notation clearly indicates that the solutions to the equation are the numbers -7 and 7. It’s important to present the solution in a clear and concise manner, especially in mathematical contexts. The solution set provides a complete answer to the problem, leaving no ambiguity about the values that make the equation true. Verifying the solutions is a crucial step in the problem-solving process. To verify that y = 7 and y = -7 are indeed solutions, we substitute each value back into the original equation, y² - 49 = 0. For y = 7, we have 7² - 49 = 49 - 49 = 0, which is true. For y = -7, we have (-7)² - 49 = 49 - 49 = 0, which is also true. This verification confirms that both values are valid solutions. The solution set {-7, 7} represents the complete and accurate answer to the equation y² - 49 = 0. Understanding how to find and express solution sets is a fundamental skill in algebra and beyond. It provides a clear and concise way to communicate the results of mathematical problem-solving.
Conclusion
In conclusion, we have successfully found the solution set for the equation y² - 49 = 0. By employing both the factoring method (difference of squares) and the square root property, we determined that the solutions are y = 7 and y = -7. The solution set, represented as {-7, 7}, encompasses all values that satisfy the given equation. This article has provided a detailed, step-by-step explanation of the solution process, ensuring a comprehensive understanding for readers. We began by understanding the nature of the equation as a quadratic equation and recognizing the difference of squares pattern. This allowed us to apply the factoring method, breaking down the equation into simpler factors and utilizing the zero-product property to find the solutions. Additionally, we explored the square root property, which offered a direct and efficient approach to solving the equation by isolating the squared variable and taking the square root of both sides. The consistency of the solutions obtained through both methods underscores the accuracy and reliability of these techniques. The ability to solve quadratic equations is a fundamental skill in mathematics, with applications spanning various fields, including physics, engineering, and computer science. Mastering different methods for solving these equations enhances problem-solving abilities and provides a deeper understanding of algebraic principles. We hope this article has been helpful in clarifying the process of solving quadratic equations and empowering you to tackle similar problems with confidence. Remember, practice is key to mastering mathematical concepts. Continue to explore and apply these techniques to a variety of equations to solidify your understanding and skills.
