Solving Quadratic Equations By Factoring 10y² - 11y - 6 = 0

Quadratic equations are a cornerstone of algebra, and mastering their solutions is essential for anyone delving into mathematics, engineering, or related fields. Among the methods for solving these equations, factorization stands out as a powerful and elegant technique. In this comprehensive guide, we will embark on a journey to unravel the intricacies of solving the quadratic equation 10y² - 11y - 6 = 0 through factorization. We'll break down the process step by step, providing clear explanations and illustrative examples to ensure a thorough understanding. Whether you're a student grappling with algebra concepts or simply seeking to enhance your mathematical toolkit, this article will equip you with the knowledge and skills to confidently tackle quadratic equations. By the end of this exploration, you'll not only be able to solve this specific equation but also grasp the underlying principles of factorization, empowering you to conquer a wide range of quadratic challenges.

Understanding Quadratic Equations

Quadratic equations, at their core, are polynomial equations of the second degree. This means they involve a variable raised to the power of two, along with other terms that may include the variable raised to the power of one and a constant term. The general form of a quadratic equation is expressed as ax² + bx + c = 0, where 'a', 'b', and 'c' represent constant coefficients, and 'x' is the variable we seek to solve for. The coefficient 'a' plays a crucial role in determining the shape and orientation of the parabola that represents the quadratic equation graphically. The coefficient 'b' influences the position of the parabola's vertex, while the constant term 'c' indicates the y-intercept of the parabola. The solutions to a quadratic equation, also known as roots or zeros, are the values of the variable 'x' that make the equation true. These solutions correspond to the points where the parabola intersects the x-axis on a graph. A quadratic equation can have up to two distinct real solutions, one real solution (a repeated root), or no real solutions, depending on the nature of the coefficients and the discriminant (b² - 4ac). The discriminant provides valuable information about the number and type of solutions, helping us to predict the outcome before even attempting to solve the equation. Understanding these fundamental aspects of quadratic equations is paramount for effectively applying various solution techniques, including factorization.

The Power of Factorization

Factorization emerges as a powerful technique for solving quadratic equations, particularly when the equation can be expressed as the product of two linear factors. This method relies on the fundamental principle that if the product of two expressions equals zero, then at least one of the expressions must be zero. In the context of quadratic equations, factorization involves transforming the quadratic expression (ax² + bx + c) into the product of two linear expressions, such as (px + q)(rx + s), where p, q, r, and s are constants. This transformation allows us to rewrite the original quadratic equation in a form that is easier to solve. Once the quadratic expression is factored, we can set each linear factor equal to zero and solve for the variable, yielding the solutions to the equation. Factorization not only provides an efficient means of solving quadratic equations but also offers insights into the structure and properties of the equation itself. By identifying the factors, we gain a deeper understanding of the roots and their relationship to the coefficients. However, it's important to recognize that not all quadratic equations can be factored easily using integers. In such cases, alternative methods like the quadratic formula or completing the square may be more appropriate. Despite its limitations, factorization remains a valuable tool in the arsenal of mathematical problem-solving techniques.

Step-by-Step Solution: Factorizing 10y² - 11y - 6 = 0

Now, let's embark on a step-by-step journey to solve the quadratic equation 10y² - 11y - 6 = 0 using the powerful technique of factorization. This process involves a series of logical steps that will guide us towards the solutions. Our first step is to identify the coefficients a, b, and c in the quadratic equation. In this case, a = 10, b = -11, and c = -6. These coefficients play a crucial role in the factorization process. The next step is to find two numbers that multiply to ac (10 * -6 = -60) and add up to b (-11). This is the heart of the factorization process, as these numbers will help us rewrite the middle term of the quadratic equation. Through careful consideration, we find that the numbers -15 and 4 satisfy these conditions (-15 * 4 = -60 and -15 + 4 = -11). Now, we rewrite the middle term (-11y) using these numbers: 10y² - 15y + 4y - 6 = 0. This step is crucial as it sets the stage for factoring by grouping. Next, we factor by grouping. We group the first two terms and the last two terms: (10y² - 15y) + (4y - 6) = 0. Then, we factor out the greatest common factor (GCF) from each group: 5y(2y - 3) + 2(2y - 3) = 0. Notice that we now have a common binomial factor (2y - 3). We factor out the common binomial factor: (2y - 3)(5y + 2) = 0. Now, we have expressed the quadratic equation as the product of two linear factors. To find the solutions, we set each factor equal to zero: 2y - 3 = 0 or 5y + 2 = 0. Solving each equation for y, we get: y = 3/2 or y = -2/5. These are the solutions to the quadratic equation 10y² - 11y - 6 = 0. We have successfully navigated the factorization process, revealing the values of y that satisfy the equation. These solutions represent the points where the parabola corresponding to the quadratic equation intersects the x-axis. The ability to solve quadratic equations through factorization empowers us to tackle a wide range of mathematical problems and applications.

Verifying the Solutions

After the factorization process, it's always a prudent step to verify the solutions we've obtained. This verification process ensures that our answers are accurate and that we haven't made any errors in our calculations. To verify the solutions, we substitute each value of y back into the original equation, 10y² - 11y - 6 = 0, and check if the equation holds true. Let's start with the first solution, y = 3/2. Substituting this value into the equation, we get: 10(3/2)² - 11(3/2) - 6 = 0. Simplifying the expression, we have: 10(9/4) - 33/2 - 6 = 0. Further simplification yields: 45/2 - 33/2 - 6 = 0. Combining the fractions, we get: 12/2 - 6 = 0, which simplifies to: 6 - 6 = 0. This confirms that y = 3/2 is indeed a valid solution. Now, let's verify the second solution, y = -2/5. Substituting this value into the equation, we get: 10(-2/5)² - 11(-2/5) - 6 = 0. Simplifying the expression, we have: 10(4/25) + 22/5 - 6 = 0. Further simplification yields: 8/5 + 22/5 - 6 = 0. Combining the fractions, we get: 30/5 - 6 = 0, which simplifies to: 6 - 6 = 0. This confirms that y = -2/5 is also a valid solution. By successfully verifying both solutions, we gain confidence in the accuracy of our factorization process and the final answers. This step is crucial in problem-solving, as it helps us catch any potential errors and ensures that we arrive at the correct conclusions. Verifying solutions is not just a formality; it's an integral part of the mathematical process that reinforces our understanding and builds confidence in our abilities.

Alternative Methods and When to Use Them

While factorization is a powerful technique for solving quadratic equations, it's not always the most efficient or straightforward method. Certain quadratic equations may prove difficult or impossible to factor using integers, necessitating the exploration of alternative solution strategies. Two prominent alternatives are the quadratic formula and completing the square. The quadratic formula, a universal solution for quadratic equations, provides a direct route to the roots regardless of the factorability of the equation. It states that for a quadratic equation of the form ax² + bx + c = 0, the solutions are given by x = [-b ± √(b² - 4ac)] / 2a. This formula guarantees a solution, even when factorization seems elusive. Completing the square, another versatile technique, involves transforming the quadratic equation into a perfect square trinomial, which can then be easily solved. This method is particularly useful when the quadratic equation cannot be factored readily and when the quadratic formula seems cumbersome. Deciding which method to employ hinges on the specific characteristics of the quadratic equation at hand. If the equation is easily factorable, factorization offers a quick and elegant solution. However, when factorization proves challenging or impossible, the quadratic formula or completing the square provide reliable alternatives. A skilled problem solver recognizes the strengths and limitations of each method, making informed decisions to navigate the quadratic landscape effectively. Mastering these alternative methods expands our mathematical toolkit, empowering us to tackle a wider array of quadratic challenges with confidence and precision.

Real-World Applications of Quadratic Equations

Quadratic equations, far from being confined to the realm of abstract mathematics, find ubiquitous applications in the real world. Their ability to model parabolic trajectories, optimization problems, and various physical phenomena makes them indispensable tools in diverse fields. In physics, quadratic equations govern the motion of projectiles, describing the path of a ball thrown through the air or the trajectory of a rocket launched into space. The height of the projectile at any given time can be precisely calculated using a quadratic equation, enabling engineers and scientists to predict and control motion with remarkable accuracy. In engineering, quadratic equations play a pivotal role in structural design. Bridges, arches, and other structures often incorporate parabolic shapes, and quadratic equations are used to determine the optimal dimensions and load-bearing capacity of these structures. By understanding the mathematical principles underlying parabolic forms, engineers can create safe and efficient designs that withstand various stresses and strains. In economics, quadratic equations are employed to model cost and revenue functions. Businesses use these models to determine the optimal pricing strategies, production levels, and investment decisions. The point at which revenue is maximized or costs are minimized can often be found by solving a quadratic equation, providing valuable insights for business planning and decision-making. Beyond these specific examples, quadratic equations permeate numerous other fields, including computer graphics, finance, and statistics. Their versatility and ability to capture non-linear relationships make them essential tools for modeling and understanding the world around us. By mastering the concepts and techniques associated with quadratic equations, we equip ourselves with a powerful lens through which to analyze and solve real-world problems.

Conclusion: Mastering Quadratic Equations

In conclusion, the journey through solving the quadratic equation 10y² - 11y - 6 = 0 by factorization has illuminated the power and elegance of this fundamental algebraic technique. We've not only unraveled the specific solutions to this equation but also delved into the broader landscape of quadratic equations, exploring their properties, alternative solution methods, and real-world applications. Factorization, with its ability to decompose a quadratic expression into the product of linear factors, provides a direct path to the roots when applicable. However, we've also recognized the importance of having alternative methods, such as the quadratic formula and completing the square, at our disposal for situations where factorization proves challenging. The ability to choose the most appropriate solution strategy based on the characteristics of the equation is a hallmark of a skilled problem solver. Beyond the mechanics of solving equations, we've emphasized the significance of verifying solutions, ensuring accuracy and building confidence in our results. The real-world applications of quadratic equations, spanning physics, engineering, economics, and beyond, underscore the practical relevance of this mathematical concept. By mastering quadratic equations, we equip ourselves with a versatile tool for modeling and solving problems in a wide range of disciplines. As we continue our mathematical journey, the knowledge and skills gained in this exploration will serve as a solid foundation for tackling more complex challenges and unlocking the deeper secrets of the mathematical universe. The journey of mathematical exploration is ongoing, and quadratic equations represent a significant milestone in our understanding of the world around us.