Solving Quadratic Equations A Step By Step Guide To Y² = 10y - 16

Introduction: Mastering Quadratic Equations

In the realm of mathematics, quadratic equations stand as fundamental tools for modeling a vast array of real-world phenomena. From the trajectory of a projectile to the optimal dimensions of a rectangular garden, these equations provide a powerful framework for understanding and solving problems. This article delves into the step-by-step solution of a specific quadratic equation: y² = 10y - 16. By exploring various methods, including factoring and the quadratic formula, we aim to equip you with the skills necessary to tackle similar problems with confidence. Understanding how to manipulate and solve quadratic equations is not just an academic exercise; it's a crucial skill for anyone venturing into fields like physics, engineering, economics, and computer science. The ability to find the roots of a quadratic equation, the values of the variable that make the equation true, is essential for optimizing designs, predicting outcomes, and making informed decisions. In this exploration, we will not only find the solutions to this particular equation but also build a solid foundation for understanding the underlying principles of quadratic equations. We'll focus on clear explanations, provide detailed steps, and emphasize the importance of checking your work to ensure accuracy. So, whether you're a student grappling with algebra or a professional seeking to refresh your mathematical toolkit, this guide offers a comprehensive approach to solving quadratic equations.

Transforming the Equation: Standard Form is Key

Before we embark on solving the equation y² = 10y - 16, it's crucial to transform it into the standard quadratic form. This form, represented as ax² + bx + c = 0, provides a structured framework that facilitates the application of various solution methods. In our case, the variable is 'y' instead of 'x', but the principle remains the same. The standard form allows us to clearly identify the coefficients a, b, and c, which are essential for both factoring and applying the quadratic formula. To convert the given equation, we need to rearrange the terms so that all terms are on one side, leaving zero on the other side. This involves subtracting 10y from both sides and adding 16 to both sides. This process maintains the equality of the equation while organizing it into a usable format. The resulting equation in standard form is y² - 10y + 16 = 0. Now, we can readily see that a = 1 (the coefficient of y²), b = -10 (the coefficient of y), and c = 16 (the constant term). These values are the building blocks for solving the equation. Understanding this transformation is paramount because it sets the stage for choosing the most efficient solution method. Whether we opt for factoring or the quadratic formula, the standard form provides a clear path forward. Moreover, this form is universally recognized, making it easier to communicate and share solutions with others. It's a cornerstone of quadratic equation solving and a skill that will serve you well in various mathematical contexts.

Method 1: Unveiling Solutions Through Factoring

Factoring is a powerful technique for solving quadratic equations, particularly when the equation can be expressed as the product of two binomials. This method relies on 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. In the context of our equation, y² - 10y + 16 = 0, we seek two numbers that multiply to 16 (the constant term) and add up to -10 (the coefficient of the y term). These numbers are -2 and -8. Recognizing this crucial pair allows us to rewrite the quadratic equation in its factored form: (y - 2)(y - 8) = 0. Now, we can apply the zero-product property. If (y - 2)(y - 8) equals zero, then either (y - 2) = 0 or (y - 8) = 0. Solving each of these simpler equations gives us the solutions. For (y - 2) = 0, we add 2 to both sides, yielding y = 2. Similarly, for (y - 8) = 0, we add 8 to both sides, resulting in y = 8. Therefore, the solutions to the equation y² - 10y + 16 = 0, obtained through factoring, are y = 2 and y = 8. Factoring is an elegant and efficient method when applicable, but it's not always straightforward. Some quadratic equations are difficult or impossible to factor using simple integer methods. However, when factoring is feasible, it provides a direct and intuitive way to find the solutions. It's a valuable tool in any mathematician's arsenal and a key concept in algebra.

Method 2: The Quadratic Formula – A Universal Solver

While factoring is a powerful tool, it's not always the most efficient or even possible method for solving quadratic equations. This is where the quadratic formula steps in as a universal solver. The quadratic formula provides a guaranteed way to find the solutions to any quadratic equation in the standard form ax² + bx + c = 0. The formula is expressed as: y = (-b ± √(b² - 4ac)) / 2a. It may seem daunting at first, but with practice, it becomes a reliable method for tackling even the most complex quadratic equations. In our case, the equation is y² - 10y + 16 = 0, where a = 1, b = -10, and c = 16. Substituting these values into the quadratic formula, we get: y = (-(-10) ± √((-10)² - 4 * 1 * 16)) / (2 * 1). Simplifying the expression, we have: y = (10 ± √(100 - 64)) / 2. Further simplification leads to: y = (10 ± √36) / 2. The square root of 36 is 6, so the equation becomes: y = (10 ± 6) / 2. Now, we have two possible solutions, one with the plus sign and one with the minus sign. For the plus sign, y = (10 + 6) / 2 = 16 / 2 = 8. For the minus sign, y = (10 - 6) / 2 = 4 / 2 = 2. Thus, the solutions obtained using the quadratic formula are y = 8 and y = 2, which match the solutions we found through factoring. The quadratic formula's strength lies in its versatility. It works regardless of whether the equation can be factored easily or not. It's a cornerstone of algebra and a skill that will empower you to solve a wide range of mathematical problems.

Verification: Ensuring Accuracy in Solutions

After finding the solutions to any equation, especially in mathematics, it's crucial to verify their accuracy. This step ensures that the values we've obtained are indeed the correct solutions and helps us avoid errors. Verification involves substituting each solution back into the original equation and checking if the equation holds true. For our equation, y² = 10y - 16, we found two solutions: y = 2 and y = 8. Let's start by verifying y = 2. Substituting y = 2 into the original equation, we get: (2)² = 10(2) - 16. This simplifies to 4 = 20 - 16, which further simplifies to 4 = 4. This is a true statement, so y = 2 is indeed a solution. Now, let's verify y = 8. Substituting y = 8 into the original equation, we get: (8)² = 10(8) - 16. This simplifies to 64 = 80 - 16, which further simplifies to 64 = 64. This is also a true statement, confirming that y = 8 is a solution as well. By verifying both solutions, we can be confident that our answers are correct. This process not only validates our work but also deepens our understanding of the equation and its solutions. Verification is an integral part of problem-solving in mathematics and should never be overlooked. It's a testament to thoroughness and a commitment to accuracy.

Conclusion: Mastering the Art of Equation Solving

In this exploration, we've successfully solved the quadratic equation y² = 10y - 16 using two distinct methods: factoring and the quadratic formula. We transformed the equation into its standard form, a crucial first step in both approaches. Factoring provided an elegant solution by expressing the equation as a product of binomials, while the quadratic formula offered a universal method, guaranteeing a solution regardless of the equation's complexity. Furthermore, we emphasized the importance of verification, a critical step in ensuring the accuracy of our solutions. By substituting the obtained values back into the original equation, we confirmed that both y = 2 and y = 8 are indeed the correct solutions. The ability to solve quadratic equations is a fundamental skill in mathematics, with applications spanning various fields. Whether you're a student tackling algebra problems or a professional applying mathematical principles in your work, a solid understanding of quadratic equations is invaluable. This article has provided a comprehensive guide, equipping you with the knowledge and techniques to confidently solve similar equations. Remember, practice is key. The more you work with quadratic equations, the more comfortable and proficient you'll become. So, embrace the challenge, explore different methods, and never underestimate the power of verification. With a solid foundation in quadratic equations, you'll be well-equipped to tackle a wide range of mathematical problems and real-world applications.