Solving Quadratic Equations A Step-by-Step Guide For Y^2 = 10y - 16

Introduction: Unraveling the Mystery of Quadratic Equations

In the realm of mathematics, quadratic equations hold a significant position, appearing in various fields such as physics, engineering, and computer science. These equations, characterized by their highest power of the variable being two, often present a fascinating challenge to solve. One such equation is $y^2 = 10y - 16$, which we will dissect and solve in this comprehensive guide. Understanding the nuances of quadratic equations and their solutions is crucial for anyone seeking to deepen their mathematical prowess.

At its core, solving a quadratic equation involves finding the values of the variable that satisfy the equation, also known as the roots or solutions. There are several methods to accomplish this, including factoring, completing the square, and the quadratic formula. Each method offers a unique approach and can be more suitable depending on the specific equation at hand. In this article, we will primarily focus on solving the equation $y^2 = 10y - 16$ by rearranging it into the standard quadratic form, factoring it, and then identifying the roots.

Before we dive into the solution, it's important to understand the significance of quadratic equations. They model a variety of real-world phenomena, from the trajectory of a projectile to the growth of populations. Mastering the techniques for solving these equations not only enhances mathematical skills but also provides a powerful tool for analyzing and understanding the world around us. This article aims to provide a clear, step-by-step guide to solving the given equation, making the process accessible to learners of all levels. We will break down each step, explaining the logic behind it, and highlighting key concepts. By the end of this guide, you will not only be able to solve $y^2 = 10y - 16$ but also gain a deeper understanding of how to approach and solve other quadratic equations.

Step 1: Rearranging the Equation into Standard Form

The first crucial step in solving the quadratic equation $y^2 = 10y - 16$ is to rearrange it into the standard quadratic form. The standard form of a quadratic equation is expressed as $ax^2 + bx + c = 0$, where a, b, and c are constants, and x is the variable. In our case, the variable is y. Rearranging the equation into this form allows us to readily apply various solving methods, such as factoring or the quadratic formula. The process of rearranging involves moving all terms to one side of the equation, leaving zero on the other side. This ensures that we have a clear and organized equation to work with.

To bring our equation, $y^2 = 10y - 16$, into the standard form, we need to subtract $10y$ and add $16$ to both sides of the equation. This maintains the equality while positioning all terms on the left side. Subtracting $10y$ from both sides gives us $y^2 - 10y = -16$. Then, adding $16$ to both sides results in $y^2 - 10y + 16 = 0$. Now, the equation is in the standard quadratic form, with $a = 1$, $b = -10$, and $c = 16$. This transformation is a critical step because it sets the stage for factoring, which is often the most straightforward method for solving quadratic equations when it is applicable.

Understanding the significance of this step is paramount. The standard form provides a clear structure that allows us to identify the coefficients and constant term, which are essential for applying factoring techniques or using the quadratic formula. Furthermore, it ensures that we are working with a consistent format, reducing the likelihood of errors in subsequent steps. By carefully rearranging the equation into standard form, we lay a solid foundation for solving it effectively and efficiently. This meticulous approach is a hallmark of mathematical problem-solving, emphasizing the importance of organization and precision.

Step 2: Factoring the Quadratic Expression

Once the equation is in standard form, $y^2 - 10y + 16 = 0$, the next step is to factor the quadratic expression. Factoring is the process of breaking down the quadratic expression into a product of two binomials. This method relies on finding two numbers that, when multiplied, give the constant term (c) and, when added, give the coefficient of the linear term (b). In our equation, we need to find two numbers that multiply to 16 and add up to -10. This is a crucial step because if we can successfully factor the quadratic expression, we can easily find the roots of the equation.

To find these numbers, we can systematically consider the factor pairs of 16. The pairs are (1, 16), (2, 8), and (4, 4). Since we need the numbers to add up to -10, and their product is positive (16), both numbers must be negative. Therefore, we consider the negative pairs: (-1, -16), (-2, -8), and (-4, -4). Among these pairs, -2 and -8 satisfy both conditions: (-2) * (-8) = 16 and (-2) + (-8) = -10. Now that we have found the numbers, we can rewrite the quadratic expression as a product of two binomials.

Using the numbers -2 and -8, we can factor the quadratic expression as follows: $y^2 - 10y + 16 = (y - 2)(y - 8)$. This factorization is a significant achievement because it transforms the quadratic equation into a product of two linear factors. Each factor represents a potential solution to the equation. The next step involves setting each factor equal to zero, which will lead us to the roots of the equation. Understanding the logic behind factoring is essential for solving quadratic equations efficiently. It provides a direct path to the solutions when the quadratic expression can be factored easily.

Step 3: Solving for y by Setting Each Factor to Zero

After successfully factoring the quadratic expression into $(y - 2)(y - 8) = 0$, the next step is to solve for y. This is achieved by applying 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 our case, this means that either $(y - 2) = 0$ or $(y - 8) = 0$, or both. By setting each factor equal to zero, we create two simple linear equations that can be easily solved to find the values of y that satisfy the original quadratic equation. This step is crucial as it directly leads us to the solutions or roots of the equation.

First, let's consider the factor $(y - 2) = 0$. To solve for y, we add 2 to both sides of the equation, which gives us $y = 2$. This is one of the solutions to the quadratic equation. Next, we consider the factor $(y - 8) = 0$. Similarly, to solve for y, we add 8 to both sides of the equation, resulting in $y = 8$. This is the second solution to the quadratic equation. Thus, we have found two values of y that make the equation $y^2 - 10y + 16 = 0$ true.

The solutions we found, $y = 2$ and $y = 8$, are the points where the parabola represented by the quadratic equation intersects the x-axis (or in this case, the y-axis since our variable is y). These values are also known as the roots or zeros of the quadratic equation. Verifying these solutions by substituting them back into the original equation, $y^2 = 10y - 16$, is a good practice to ensure accuracy. For $y = 2$, we have $2^2 = 10(2) - 16$, which simplifies to $4 = 20 - 16$, or $4 = 4$, confirming the solution. For $y = 8$, we have $8^2 = 10(8) - 16$, which simplifies to $64 = 80 - 16$, or $64 = 64$, also confirming the solution. This step of solving for y and verifying the solutions solidifies our understanding of the quadratic equation and its roots.

Step 4: Verifying the Solutions

After obtaining the solutions $y = 2$ and $y = 8$, it is essential to verify these solutions to ensure their accuracy. Verification involves substituting each solution back into the original equation, $y^2 = 10y - 16$, to check if the equation holds true. This step is a critical part of the problem-solving process as it helps to identify any errors that may have occurred during the earlier steps. By confirming that the solutions satisfy the original equation, we can be confident in the correctness of our answer. This practice not only validates the solutions but also reinforces the understanding of the equation and its properties.

Let's start by verifying the solution $y = 2$. Substituting $y = 2$ into the original equation gives us: $(2)^2 = 10(2) - 16$. Simplifying the equation, we get $4 = 20 - 16$, which further simplifies to $4 = 4$. This equality confirms that $y = 2$ is indeed a valid solution to the equation. Now, let's verify the second solution, $y = 8$. Substituting $y = 8$ into the original equation gives us: $(8)^2 = 10(8) - 16$. Simplifying this equation, we get $64 = 80 - 16$, which simplifies to $64 = 64$. This equality confirms that $y = 8$ is also a valid solution to the equation.

The process of verification not only confirms the solutions but also provides a deeper understanding of the equation's behavior. By substituting the solutions back into the original equation, we can see how the values of y relate to the equation's structure and the equality it represents. This step is particularly important in mathematics as it emphasizes the importance of checking one's work and ensuring the accuracy of the results. Furthermore, it highlights the interconnectedness of different mathematical concepts, such as substitution and equation solving. In conclusion, verifying the solutions is a crucial step in solving any equation, as it provides a final check and reinforces the understanding of the mathematical principles involved.

Conclusion: Mastering Quadratic Equations

In conclusion, we have successfully solved the quadratic equation $y^2 = 10y - 16$ by following a systematic approach. We began by rearranging the equation into the standard quadratic form, $y^2 - 10y + 16 = 0$. This step was crucial as it allowed us to identify the coefficients and constant term, which are essential for applying factoring techniques. Next, we factored the quadratic expression into $(y - 2)(y - 8) = 0$. Factoring transformed the equation into a product of two linear factors, making it easier to find the solutions. Then, we applied the zero-product property, setting each factor equal to zero and solving for y, which gave us the solutions $y = 2$ and $y = 8$. Finally, we verified these solutions by substituting them back into the original equation, ensuring their accuracy. This step-by-step process demonstrates a comprehensive approach to solving quadratic equations, highlighting the importance of each step in achieving the correct solutions.

Mastering quadratic equations is a fundamental skill in mathematics, with applications spanning various fields. The techniques used in solving $y^2 = 10y - 16$ can be applied to a wide range of quadratic equations, providing a solid foundation for more advanced mathematical concepts. The ability to rearrange equations into standard form, factor quadratic expressions, and apply the zero-product property are essential tools in any mathematician's toolkit. Furthermore, the emphasis on verifying solutions underscores the importance of accuracy and attention to detail in mathematical problem-solving. This methodical approach not only ensures the correctness of the solutions but also deepens the understanding of the underlying mathematical principles.

The journey through solving this quadratic equation has provided valuable insights into the nature of these equations and the methods used to solve them. By breaking down the process into manageable steps, we have demonstrated that even seemingly complex equations can be tackled with a clear and systematic approach. This article serves as a testament to the power of mathematical problem-solving and the satisfaction that comes from successfully navigating a challenging equation. With a solid understanding of the principles and techniques outlined in this guide, readers are well-equipped to tackle future quadratic equations and further explore the fascinating world of mathematics.