Solving 2x² - 11x + 5 = 0: Methods And Step-by-Step Guide

Introduction

Hey guys! Today, let's dive into solving a quadratic equation. Quadratic equations are a fundamental topic in algebra, and mastering them is super important for more advanced math. We're going to tackle the equation 2x² - 11x + 5 = 0 step-by-step. Don't worry if it looks intimidating – we'll break it down into easy-to-understand chunks. We'll explore different methods, including factoring, using the quadratic formula, and even completing the square. By the end of this guide, you'll not only know how to solve this specific equation but also have a solid understanding of how to approach similar problems. Whether you're a student prepping for an exam or just a math enthusiast, this guide is for you. So, let's get started and unlock the secrets of quadratic equations!

In this comprehensive guide, we will meticulously dissect the quadratic equation 2x² - 11x + 5 = 0, providing a detailed walkthrough of multiple solution methods. Quadratic equations, which take the general form of ax² + bx + c = 0, where a, b, and c are constants, are a cornerstone of algebra. Understanding how to solve them opens the door to a wide range of mathematical applications, from physics and engineering to economics and computer science. We'll begin by exploring the factoring method, a classic technique that relies on breaking down the quadratic expression into simpler factors. This method is particularly efficient when the equation can be easily factored, allowing us to quickly identify the roots. Next, we'll delve into the quadratic formula, a universally applicable method that provides solutions for any quadratic equation, regardless of its factorability. This formula is a powerful tool in any mathematician's arsenal and ensures we can always find the roots, even when factoring proves challenging. We'll also touch upon completing the square, a technique that transforms the quadratic equation into a perfect square trinomial, making it easier to solve. By understanding these different methods, you'll gain a versatile toolkit for tackling quadratic equations and a deeper appreciation for the underlying principles of algebra. Each method will be explained with clear steps and examples, ensuring you can confidently apply them to various problems. So, let's embark on this mathematical journey together and master the art of solving quadratic equations!

We'll kick things off by understanding what a quadratic equation really is. Simply put, a quadratic equation is a polynomial equation of the second degree. That means the highest power of the variable (in our case, x) 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. If a were zero, the equation would become linear, not quadratic. In our example, 2x² - 11x + 5 = 0, we can identify a as 2, b as -11, and c as 5. These coefficients play a crucial role in determining the solutions, or roots, of the equation. Understanding the coefficients helps us choose the most appropriate method for solving the equation. For instance, the magnitude and signs of a, b, and c can give us clues about whether the equation is easily factorable or if we should resort to the quadratic formula. Visualizing a quadratic equation as a parabola can also provide valuable insights. The roots of the equation correspond to the x-intercepts of the parabola, which are the points where the parabola crosses the x-axis. The shape and position of the parabola are determined by the coefficients a, b, and c. If the parabola opens upwards (a > 0), it has a minimum point, and if it opens downwards (a < 0), it has a maximum point. Understanding these visual and algebraic connections helps to solidify our grasp of quadratic equations and their properties. So, before we jump into solving, let's appreciate the structure and nature of these equations, which will make our problem-solving journey much smoother.

Method 1: Factoring

Alright, let's start with the factoring method. Factoring is a fantastic way to solve quadratic equations if you can spot the factors easily. The main idea behind factoring is to rewrite the quadratic equation as a product of two binomials. So, we want to express 2x² - 11x + 5 = 0 in the form (px + q)(rx + s) = 0. To do this, we need to find two numbers that multiply to give the product of the leading coefficient (2) and the constant term (5), which is 2 * 5 = 10, and add up to the middle coefficient (-11). Take a moment to think about those numbers... They are -10 and -1! Now, we rewrite the middle term (-11x) using these numbers: 2x² - 10x - x + 5 = 0. Next, we factor by grouping. We group the first two terms and the last two terms: (2x² - 10x) + (-x + 5) = 0. Now, we factor out the greatest common factor from each group. From the first group, we can factor out 2x, and from the second group, we can factor out -1: 2x(x - 5) - 1(x - 5) = 0. Notice that we now have a common factor of (x - 5). We factor this out: (x - 5)(2x - 1) = 0. Finally, we set each factor equal to zero and solve for x: x - 5 = 0 or 2x - 1 = 0. Solving these linear equations, we get x = 5 or x = 1/2. So, the solutions to the equation 2x² - 11x + 5 = 0 are x = 5 and x = 1/2. See? Factoring can be pretty straightforward once you get the hang of it! Remember to always double-check your factors and your solutions to make sure they're correct.

Factoring is a powerful technique for solving quadratic equations, but it's most effective when the equation can be easily factored. It involves expressing the quadratic equation in the form ax² + bx + c = 0 as a product of two binomials, such as (px + q)(rx + s) = 0. The key to mastering factoring lies in finding the correct combination of numbers that satisfy the required conditions. For the equation 2x² - 11x + 5 = 0, we need to identify two numbers that multiply to the product of the leading coefficient (2) and the constant term (5), which equals 10, and simultaneously add up to the middle coefficient (-11). This step often requires a bit of trial and error, but with practice, you'll develop an intuition for identifying these numbers quickly. Once we've found the correct numbers (-10 and -1 in this case), we rewrite the middle term (-11x) as a sum of these numbers multiplied by x, resulting in 2x² - 10x - x + 5 = 0. This step is crucial as it allows us to group terms and factor by grouping. Factoring by grouping involves pairing the first two terms and the last two terms together: (2x² - 10x) + (-x + 5) = 0. We then factor out the greatest common factor (GCF) from each group. From the first group, the GCF is 2x, and from the second group, the GCF is -1. This gives us 2x(x - 5) - 1(x - 5) = 0. Notice that we now have a common binomial factor of (x - 5). We factor this out, resulting in (x - 5)(2x - 1) = 0. This is the factored form of the quadratic equation. To find the solutions, we set each factor equal to zero: x - 5 = 0 and 2x - 1 = 0. Solving these simple linear equations yields the solutions x = 5 and x = 1/2. These are the roots of the quadratic equation. Factoring is an elegant and efficient method, but it's important to remember that not all quadratic equations can be easily factored. In such cases, we turn to other methods like the quadratic formula or completing the square.

Let's dig deeper into the nuances of factoring to make sure we've got a solid grasp of this method. One of the most common hurdles students face is identifying the correct numbers that both multiply to ac and add up to b, where a, b, and c are the coefficients in the quadratic equation ax² + bx + c = 0. In our example, 2x² - 11x + 5 = 0, this means finding two numbers that multiply to 2 * 5 = 10 and add up to -11. It's often helpful to systematically list out factor pairs of 10 (1 and 10, 2 and 5) and consider their signs to see which pair sums to -11. Remember, both numbers must be negative since their product is positive (10) and their sum is negative (-11). Another critical step is the factoring by grouping process. After rewriting the middle term, we group the terms and factor out the greatest common factor (GCF) from each group. This step is crucial for simplifying the expression and revealing the common binomial factor. Make sure you factor out the correct GCF from each group, including the sign. A common mistake is to forget the negative sign when factoring from the second group, which can lead to incorrect factorization. Once you've factored out the GCFs, you should see a common binomial factor emerge. This is a good sign that you're on the right track. Factoring out the common binomial factor is the final step in expressing the quadratic equation as a product of two binomials. From there, setting each factor to zero is straightforward and leads to the solutions. Always double-check your solutions by substituting them back into the original equation to ensure they satisfy it. This is a simple yet effective way to catch any errors you might have made during the factoring process. Factoring is not just a mechanical process; it requires a deep understanding of the relationships between the coefficients and factors of a quadratic equation. With practice, you'll develop a knack for spotting factorable equations and efficiently finding their solutions. And remember, if factoring seems too difficult or time-consuming, the quadratic formula is always a reliable alternative.

Method 2: Quadratic Formula

Now, let's talk about the quadratic formula. Guys, this formula is your best friend when factoring gets tough or seems impossible. The quadratic formula is a universal solution for any quadratic equation in the form ax² + bx + c = 0. It states that the solutions for x are given by: x = (-b ± √(b² - 4ac)) / (2a). Don't let the formula intimidate you – it's actually quite straightforward once you get the hang of plugging in the values. For our equation, 2x² - 11x + 5 = 0, we know that a = 2, b = -11, and c = 5. Let's substitute these values into the formula: x = (-(-11) ± √((-11)² - 4 * 2 * 5)) / (2 * 2). Simplify step by step: x = (11 ± √(121 - 40)) / 4. Continue simplifying: x = (11 ± √81) / 4. Since the square root of 81 is 9, we have: x = (11 ± 9) / 4. Now, we have two possible solutions: x = (11 + 9) / 4 or x = (11 - 9) / 4. For the first solution: x = 20 / 4 = 5. For the second solution: x = 2 / 4 = 1/2. So, using the quadratic formula, we again find that the solutions are x = 5 and x = 1/2. See? The quadratic formula always works, even when factoring is tricky. It's a reliable tool in your math arsenal, so make sure you're comfortable using it! Remember, the key is to correctly identify a, b, and c, and then carefully substitute them into the formula and simplify. With practice, you'll become a pro at using the quadratic formula.

The quadratic formula is a fundamental tool in solving quadratic equations, providing a guaranteed method to find the roots regardless of whether the equation is easily factorable. This formula is derived from the method of completing the square and is universally applicable to any quadratic equation in the standard form ax² + bx + c = 0. The formula itself is expressed as: x = (-b ± √(b² - 4ac)) / (2a). It's crucial to memorize this formula as it's a cornerstone of algebra and will be used extensively in various mathematical contexts. To apply the quadratic formula, the first step is to correctly identify the coefficients a, b, and c from the given equation. In our example, 2x² - 11x + 5 = 0, we have a = 2, b = -11, and c = 5. The next step is to carefully substitute these values into the formula. Pay close attention to the signs, as incorrect substitution is a common source of errors. After substitution, we have: x = (-(-11) ± √((-11)² - 4 * 2 * 5)) / (2 * 2). The next step involves simplifying the expression. Start by simplifying the terms inside the square root. The discriminant, b² - 4ac, plays a critical role in determining the nature of the roots. If the discriminant is positive, the equation has two distinct real roots. If it's zero, the equation has one real root (a repeated root). If it's negative, the equation has two complex roots. In our case, (-11)² - 4 * 2 * 5 = 121 - 40 = 81. Since 81 is positive, we expect two distinct real roots. Continuing with the simplification, we have x = (11 ± √81) / 4. The square root of 81 is 9, so we get x = (11 ± 9) / 4. This gives us two possible solutions: x = (11 + 9) / 4 and x = (11 - 9) / 4. Solving these, we get x = 20 / 4 = 5 and x = 2 / 4 = 1/2. These are the roots of the quadratic equation, consistent with our solution obtained through factoring. The quadratic formula is a powerful tool because it provides a systematic approach to solving quadratic equations, regardless of their complexity or factorability. It's a must-have in your mathematical toolkit.

Let's break down the quadratic formula even further to ensure you're completely comfortable using it. One of the most critical aspects of the quadratic formula is the discriminant, which is the expression b² - 4ac under the square root. The discriminant provides valuable information about the nature of the roots of the quadratic equation. As we mentioned earlier, if the discriminant is positive, the equation has two distinct real roots, which means the parabola intersects the x-axis at two different points. If the discriminant is zero, the equation has one real root (a repeated root), which means the parabola touches the x-axis at exactly one point. If the discriminant is negative, the equation has two complex roots, which means the parabola does not intersect the x-axis. Understanding the discriminant can help you predict the type of solutions you'll get before you even solve the equation. This can be particularly useful in problem-solving scenarios where you need to quickly determine the nature of the roots without finding their exact values. Another key point is the proper substitution of the coefficients a, b, and c into the formula. Pay close attention to the signs, especially the negative signs. A common mistake is to forget the negative sign in front of b in the formula or to incorrectly square a negative number. Always double-check your substitutions to avoid these errors. When simplifying the expression, follow the order of operations (PEMDAS/BODMAS) carefully. First, simplify the expression under the square root, then take the square root, and finally perform the addition and subtraction. Remember that the ± symbol indicates that there are two possible solutions, one with the plus sign and one with the minus sign. After you've found the solutions, it's always a good idea to check them by substituting them back into the original equation. This will help you catch any errors you might have made during the process. The quadratic formula is a powerful and versatile tool, but it requires careful attention to detail and a solid understanding of algebraic principles. With practice and patience, you'll master this formula and be able to confidently solve any quadratic equation. So, don't be intimidated by it; embrace it as a reliable ally in your mathematical journey!

Conclusion

So, there you have it! We've successfully solved the quadratic equation 2x² - 11x + 5 = 0 using both factoring and the quadratic formula. We found that the solutions are x = 5 and x = 1/2. Remember, guys, the method you choose depends on the equation. If you can easily factor the equation, that's often the quickest route. But if factoring seems tricky, the quadratic formula is always a reliable option. Mastering these methods will give you a solid foundation for tackling more complex math problems in the future. Keep practicing, and you'll become a quadratic equation-solving pro in no time! Quadratic equations are a fundamental part of algebra, and understanding them opens the door to many other areas of mathematics and science. So, keep up the great work, and happy solving!

In conclusion, we have thoroughly explored the solution of the quadratic equation 2x² - 11x + 5 = 0, employing two primary methods: factoring and the quadratic formula. Through factoring, we skillfully decomposed the quadratic expression into the product of two binomials, leading us to the solutions x = 5 and x = 1/2. This method, while efficient when applicable, requires a keen eye for identifying the correct factors. We then demonstrated the versatility of the quadratic formula, a universal solution that ensures we can find the roots of any quadratic equation, regardless of its factorability. By substituting the coefficients a = 2, b = -11, and c = 5 into the formula, we arrived at the same solutions: x = 5 and x = 1/2. This reaffirms the reliability and power of the quadratic formula as a problem-solving tool. The choice between factoring and the quadratic formula often depends on the specific equation at hand. If the equation is easily factorable, factoring may be the quicker route. However, the quadratic formula serves as a steadfast backup, guaranteeing a solution even when factoring proves challenging. Mastering both methods provides a comprehensive approach to tackling quadratic equations and enhances your mathematical toolkit. These skills are not only essential for academic pursuits but also find applications in various fields, including physics, engineering, and economics. Quadratic equations are a cornerstone of mathematical education, and a solid understanding of their solution methods is crucial for success in higher-level mathematics. So, continue practicing these techniques, and you'll be well-equipped to tackle any quadratic equation that comes your way. Keep exploring the world of mathematics, and you'll discover its beauty and power in solving real-world problems.

To further solidify your understanding of quadratic equations and their solutions, consider exploring additional resources and practice problems. There are numerous online platforms and textbooks that offer a wide range of exercises, from simple factorable equations to more complex problems that require the quadratic formula. Working through these exercises will help you develop your problem-solving skills and gain confidence in your ability to tackle any quadratic equation. In addition to practice problems, it's also beneficial to delve deeper into the theory behind quadratic equations. Understanding the derivation of the quadratic formula, the relationship between the coefficients and the roots, and the graphical representation of quadratic functions (parabolas) will provide a more holistic understanding of the topic. You can also explore applications of quadratic equations in real-world scenarios. For example, quadratic equations are used in physics to model projectile motion, in engineering to design bridges and structures, and in economics to analyze supply and demand curves. Seeing these real-world applications can make the topic more engaging and help you appreciate its relevance. Remember, mastering quadratic equations is not just about memorizing formulas and procedures; it's about developing a deep understanding of the underlying concepts and principles. So, take the time to explore, experiment, and ask questions. The more you engage with the material, the more proficient you'll become in solving quadratic equations and applying them to various problems. And most importantly, don't be afraid to make mistakes. Mistakes are a natural part of the learning process, and they provide valuable opportunities for growth and improvement. So, embrace the challenges, persevere through the difficulties, and celebrate your successes along the way. Happy learning!

Solve the equation 2x² - 11x + 5 = 0.

Solving 2x² - 11x + 5 = 0: Methods and Step-by-Step Guide