Solution Set Of Y² = 100: A Step-by-Step Guide
This article provides a detailed explanation of how to find the solution set for the equation y² = 100. We'll explore the fundamental concepts of quadratic equations, delve into the methods for solving them, and ultimately arrive at the correct solution set. Understanding how to solve such equations is crucial in various areas of mathematics and its applications, making this a valuable skill to acquire. This exploration isn't just about getting the right answer; it's about grasping the underlying principles that govern these types of problems. We'll break down each step, ensuring clarity and building a strong foundation for more advanced mathematical concepts. Whether you're a student grappling with algebra or simply someone looking to refresh their math skills, this guide offers a comprehensive and accessible approach to solving this equation.
Understanding Quadratic Equations
Before diving into the specific solution, let's first understand the broader concept of quadratic equations. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable is 2. The general form of a quadratic equation is ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. Our equation, y² = 100, is a special case of a quadratic equation where b = 0 and we can rewrite it as y² - 100 = 0. Recognizing this form is crucial because it allows us to apply specific techniques for solving quadratic equations. The solutions to a quadratic equation are also known as its roots or zeros, representing the values of the variable that make the equation true. These roots have significant implications in various mathematical contexts, including graphing parabolas and solving optimization problems. In essence, quadratic equations are fundamental building blocks in algebra, and mastering their solutions opens doors to a wider understanding of mathematical concepts.
Methods for Solving y² = 100
There are several methods to solve the equation y² = 100. We'll explore two primary approaches: the square root method and factoring. The square root method is particularly efficient for equations in the form x² = k, where k is a constant. This method involves taking the square root of both sides of the equation, remembering to consider both positive and negative roots. In our case, taking the square root of both sides of y² = 100 gives us y = ±√100, which simplifies to y = ±10. The second method, factoring, involves rewriting the equation in the form (y - a)(y - b) = 0, where 'a' and 'b' are the roots of the equation. We can rewrite y² - 100 = 0 as (y - 10)(y + 10) = 0. Setting each factor equal to zero gives us y - 10 = 0 and y + 10 = 0, which leads to the solutions y = 10 and y = -10. Both methods demonstrate how different approaches can lead to the same solution, reinforcing the importance of understanding multiple problem-solving techniques. Choosing the most appropriate method often depends on the specific form of the equation, but having a repertoire of strategies allows for greater flexibility and efficiency.
Applying the Square Root Method
The square root method is a direct and efficient way to solve equations in the form y² = k. The underlying principle is that if two quantities are equal, their square roots must also be equal. However, a crucial point to remember is that every positive number has two square roots: a positive square root and a negative square root. This is because both the positive and negative values, when squared, will result in the original positive number. For instance, both 10² and (-10)² equal 100. Applying this to our equation, y² = 100, we take the square root of both sides: √(y²) = ±√100. This gives us y = ±10. This notation, ±10, signifies that there are two possible solutions: y = 10 and y = -10. Failing to consider both positive and negative roots is a common mistake, so it's crucial to explicitly include the ± sign when applying the square root method. Understanding this principle ensures that we capture all possible solutions to the equation. The square root method provides a clear and concise pathway to solving equations of this type, making it a valuable tool in any mathematical toolkit.
Applying Factoring
Factoring is another powerful method for solving quadratic equations. It involves rewriting the equation as a product of two binomials. To apply factoring to y² = 100, we first rewrite the equation in the standard quadratic form: y² - 100 = 0. This equation is a difference of squares, which can be factored using the formula a² - b² = (a - b)(a + b). In our case, a = y and b = 10, so we can factor y² - 100 as (y - 10)(y + 10). Now, we have the equation (y - 10)(y + 10) = 0. The principle behind solving by factoring is that if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we set each factor equal to zero: y - 10 = 0 and y + 10 = 0. Solving these linear equations gives us y = 10 and y = -10. This method highlights the relationship between the roots of a quadratic equation and its factored form. Factoring provides an alternative approach to the square root method, showcasing the versatility of algebraic techniques. Understanding factoring is not only useful for solving equations but also for simplifying expressions and analyzing polynomial functions.
The Solution Set
After applying either the square root method or factoring, we arrive at the same two solutions for the equation y² = 100: y = 10 and y = -10. The solution set is the set of all values that satisfy the equation. In this case, the solution set is {10, -10}. It's important to express the solution in set notation to clearly indicate that these are the values that make the equation true. The solution set provides a concise representation of all possible solutions, ensuring clarity and accuracy in mathematical communication. We can verify these solutions by substituting them back into the original equation. If we substitute y = 10, we get 10² = 100, which is true. If we substitute y = -10, we get (-10)² = 100, which is also true. This verification step confirms the accuracy of our solutions. The solution set encapsulates the final answer, providing a complete and unambiguous resolution to the problem. Understanding how to determine and express the solution set is a fundamental aspect of solving equations in mathematics.
Verification and Conclusion
To ensure the accuracy of our solutions, we verify them by substituting them back into the original equation. Substituting y = 10 into y² = 100 gives us 10² = 100, which simplifies to 100 = 100, a true statement. Similarly, substituting y = -10 into y² = 100 gives us (-10)² = 100, which simplifies to 100 = 100, also a true statement. This verification process confirms that both y = 10 and y = -10 are indeed solutions to the equation. In conclusion, the solution set for the equation y² = 100 is {10, -10}. We arrived at this solution by applying two different methods: the square root method and factoring. Both methods demonstrate the importance of understanding fundamental algebraic principles and applying them correctly. This comprehensive guide has not only provided the solution but has also delved into the underlying concepts and techniques, equipping you with the knowledge to solve similar problems in the future. Mastering these skills is crucial for success in algebra and beyond, paving the way for a deeper understanding of mathematics.