Solving The Quadratic Equation (5x+4)(x+1)=2 A Step-by-Step Guide

Introduction

In the realm of algebra, solving equations stands as a fundamental skill, enabling us to unravel the mysteries behind mathematical relationships. Among the various types of equations, quadratic equations hold a significant place, appearing in diverse fields ranging from physics and engineering to economics and computer science. This article delves into the intricacies of solving the quadratic equation $(5x + 4)(x + 1) = 2$, employing a systematic approach that empowers you to tackle similar challenges with confidence. We will explore the equation $(5x + 4)(x + 1) = 2$ step by step, highlighting key concepts and techniques along the way. This comprehensive guide will equip you with the necessary tools to navigate the world of quadratic equations and emerge victorious. Understanding quadratic equations is crucial for anyone venturing into advanced mathematics or applied sciences. The ability to solve these equations unlocks a deeper understanding of mathematical principles and their real-world applications. Whether you're a student preparing for an exam or a professional seeking to enhance your problem-solving skills, mastering quadratic equations is an invaluable asset. This article will serve as your compass, guiding you through the intricacies of this mathematical domain and empowering you to excel. As we embark on this journey, remember that practice is the key to mastery. The more you engage with quadratic equations, the more comfortable and confident you will become in your ability to solve them. So, let's dive in and unlock the secrets of quadratic equations together. This journey starts with a seemingly simple equation, but it opens the door to a vast and fascinating world of mathematical possibilities. The ability to manipulate and solve equations like this is not just an academic exercise; it's a skill that translates to real-world problem-solving. From calculating projectile trajectories to optimizing business decisions, the principles we'll explore here have far-reaching implications. So, prepare to sharpen your algebraic acumen and embark on a journey of mathematical discovery.

Expanding and Simplifying the Equation

Our initial task involves transforming the given equation into a standard quadratic form, which is essential for applying various solution methods. The equation we're tackling is $(5x + 4)(x + 1) = 2$. The first step is to expand the left side of the equation by applying the distributive property (often referred to as the FOIL method). This involves multiplying each term in the first parenthesis by each term in the second parenthesis. So, let's break it down: $(5x + 4)(x + 1) = 5x * x + 5x * 1 + 4 * x + 4 * 1$. This simplifies to $5x^2 + 5x + 4x + 4$. Now, we combine the like terms ($5x$ and $4x$) to get $5x^2 + 9x + 4$. Remember, the goal here is to get the equation into a form that we can easily work with. This expansion is a crucial step in solving quadratic equations, as it allows us to identify the coefficients that will be used in subsequent steps, such as applying the quadratic formula or completing the square. By carefully expanding and simplifying, we lay the groundwork for a successful solution. Now we have: $5x^2 + 9x + 4 = 2$. To achieve the standard quadratic form ($ax^2 + bx + c = 0$), we need to move the constant term from the right side to the left side of the equation. This is done by subtracting 2 from both sides of the equation. So, $5x^2 + 9x + 4 - 2 = 2 - 2$, which simplifies to $5x^2 + 9x + 2 = 0$. This equation is now in the standard quadratic form, where $a = 5$, $b = 9$, and $c = 2$. This form is essential for applying various methods to solve the equation, including factoring, using the quadratic formula, or completing the square. The process of expanding and simplifying is not just about manipulating symbols; it's about transforming the equation into a more manageable representation. This transformation allows us to see the underlying structure of the equation and apply appropriate solution techniques. With the equation now in standard form, we are well-positioned to explore different methods for finding the values of x that satisfy the equation. This careful attention to detail in the initial steps will pay dividends as we move towards the solution.

Solving by Factoring

Factoring is a powerful technique for solving quadratic equations, especially when the equation can be expressed as a 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. Our equation, now in standard form, is $5x^2 + 9x + 2 = 0$. To factor this equation, we need to find two binomials that, when multiplied, give us the original quadratic expression. This often involves a bit of trial and error, but there are systematic approaches we can use. First, we look for two numbers that multiply to give us the product of the leading coefficient (5) and the constant term (2), which is 10. At the same time, these two numbers must add up to the middle coefficient (9). After some consideration, we find that the numbers 1 and 10 satisfy these conditions (1 * 10 = 10 and 1 + 10 = 11). But 1 and 8 will not work. Instead we have the numbers 1 and 8 won't do the job, we look for factors that satisfy both conditions. Now, we rewrite the middle term (9x) using these two numbers. So, $5x^2 + 9x + 2$ becomes $5x^2 + x + 8x + 2$ This step is crucial as it allows us to group terms and factor by grouping. Next, we factor by grouping. We group the first two terms and the last two terms: $(5x^2 + x) + (8x + 2)$. From the first group, we can factor out an x, giving us $x(5x + 1)$. From the second group, we can factor out a 2, giving us $2(4x + 1)$. Now, our expression looks like this: $x(5x + 1) + 2(4x + 1)$. Notice that the terms inside the parentheses are not the same, indicating that we may have made an error in our factoring process or that this equation might not be easily factorable. If factoring proves difficult, we can always turn to other methods like the quadratic formula. However, let's re-examine our steps to ensure we haven't overlooked anything. Factoring is not always straightforward, and it requires careful attention to detail. The key is to find the right combination of factors that will lead to the correct binomial expressions. If we are unable to find such factors, it might be a sign that the quadratic formula is a more suitable approach for this particular equation. In the next section, we will explore the quadratic formula and demonstrate how it can be used to solve equations that are not easily factored. This versatility is what makes the quadratic formula such a valuable tool in algebra.

Applying the Quadratic Formula

When factoring proves challenging or impossible, the quadratic formula emerges as a reliable and universally applicable method for solving quadratic equations. This formula provides a direct solution for the roots of any quadratic equation in the standard form $ax^2 + bx + c = 0$. The quadratic formula is expressed as: $x = (-b ± √(b^2 - 4ac)) / (2a)$. This formula might look intimidating at first, but with practice, it becomes a powerful tool in your problem-solving arsenal. To apply the quadratic formula, we first need to identify the coefficients a, b, and c from our equation, which is $5x^2 + 9x + 2 = 0$. In this case, $a = 5$, $b = 9$, and $c = 2$. Now, we simply substitute these values into the formula. Let's break it down step by step: First, we calculate the discriminant, which is the part under the square root: $b^2 - 4ac$. Substituting the values, we get $9^2 - 4 * 5 * 2 = 81 - 40 = 41$. The discriminant tells us about the nature of the roots. If it's positive, we have two distinct real roots; if it's zero, we have one real root (a repeated root); and if it's negative, we have two complex roots. In our case, the discriminant is 41, which is positive, indicating that we have two distinct real roots. Next, we substitute the values of a, b, and the discriminant into the quadratic formula: $x = (-9 ± √41) / (2 * 5)$. This simplifies to $x = (-9 ± √41) / 10$. Now, we have two possible solutions for x, one with the plus sign and one with the minus sign. Let's calculate them separately: $x_1 = (-9 + √41) / 10$ and $x_2 = (-9 - √41) / 10$. These are the exact solutions to the quadratic equation. We can approximate these values using a calculator if we need decimal representations. The quadratic formula is a cornerstone of algebra, and mastering its application is crucial for solving a wide range of problems. It's a testament to the power of mathematical formulas to provide solutions to complex equations. With the quadratic formula in your toolkit, you can confidently tackle any quadratic equation that comes your way. Remember, practice is key. The more you use the formula, the more comfortable and proficient you will become in its application. This confidence will serve you well in your mathematical endeavors.

Solutions and Verification

Having applied the quadratic formula, we have obtained two potential solutions for our equation, $5x^2 + 9x + 2 = 0$. These solutions are $x_1 = (-9 + √41) / 10$ and $x_2 = (-9 - √41) / 10$. To ensure the accuracy of our solutions, it's essential to verify them by substituting them back into the original equation. This process not only confirms that we have solved the equation correctly but also reinforces our understanding of the relationship between the roots and the equation. Let's start by verifying the first solution, $x_1 = (-9 + √41) / 10$. We substitute this value into the original equation, $(5x + 4)(x + 1) = 2$. This involves a bit of algebraic manipulation, but by carefully following the order of operations, we can determine whether the equation holds true. Similarly, we verify the second solution, $x_2 = (-9 - √41) / 10$, by substituting it into the original equation. Again, we perform the necessary calculations to see if the equation is satisfied. If both solutions satisfy the original equation, we can confidently conclude that we have found the correct roots. Verification is a critical step in the problem-solving process, especially in mathematics. It's a way of double-checking our work and ensuring that our solutions are valid. This practice not only enhances our accuracy but also deepens our understanding of the concepts involved. In this case, verifying the solutions to the quadratic equation reinforces our understanding of the relationship between the roots and the equation itself. It's a testament to the importance of precision and attention to detail in mathematics. By taking the time to verify our solutions, we demonstrate a commitment to accuracy and a desire to fully grasp the underlying principles. This approach will serve you well in all your mathematical pursuits. Now that we have verified our solutions, we can present them with confidence. These solutions represent the values of x that make the equation true, and they are the culmination of our efforts in applying the quadratic formula. The process of solving and verifying quadratic equations is a journey that strengthens our algebraic skills and builds our problem-solving abilities. With each equation we solve, we gain a deeper appreciation for the elegance and power of mathematics.

Conclusion

In this comprehensive guide, we have successfully navigated the process of solving the quadratic equation $(5x + 4)(x + 1) = 2$. We began by expanding and simplifying the equation to its standard form, $5x^2 + 9x + 2 = 0$. We then explored the method of factoring, which, while not directly applicable in this case, provided valuable insights into the structure of quadratic equations. The quadratic formula emerged as our primary tool, offering a reliable and universally applicable method for finding the roots of the equation. We carefully applied the formula, identifying the coefficients a, b, and c, and substituting them into the formula. This led us to the solutions $x_1 = (-9 + √41) / 10$ and $x_2 = (-9 - √41) / 10$. To ensure the accuracy of our results, we meticulously verified these solutions by substituting them back into the original equation. This process confirmed that our solutions were indeed correct, reinforcing our understanding of the relationship between the roots and the equation. This journey through solving a quadratic equation has highlighted several key concepts and skills. We have reinforced the importance of algebraic manipulation, the power of the quadratic formula, and the necessity of verification in problem-solving. These skills are not only valuable in mathematics but also in various fields that require analytical thinking and problem-solving abilities. Quadratic equations are fundamental to many areas of science, engineering, and economics. Understanding how to solve them opens doors to a deeper understanding of these fields and the ability to tackle real-world problems. This article has provided a solid foundation for further exploration of quadratic equations and other algebraic concepts. By mastering the techniques presented here, you will be well-equipped to face more complex mathematical challenges. Remember, the key to success in mathematics is practice and perseverance. The more you engage with problems and apply the concepts you have learned, the more confident and proficient you will become. This journey of learning is a continuous process, and each step you take builds upon the previous one. So, continue to explore, question, and practice, and you will unlock the vast and fascinating world of mathematics. The ability to solve equations is a powerful tool, and it is one that will serve you well throughout your academic and professional life. Embrace the challenge, and enjoy the journey of mathematical discovery.