Solving 6x² - 30x - 84 = 0 Step-by-Step Guide
Hey guys! Today, we're diving into the exciting world of quadratic equations and tackling a specific problem: finding the x values that satisfy the equation 6x² - 30x - 84 = 0. Don't worry if it looks intimidating at first glance; we'll break it down step by step and make it super easy to understand. We will explore different methods to solve this equation, providing a comprehensive guide for anyone looking to master quadratic equations. So, let's put on our math hats and get started!
Understanding Quadratic Equations
Before we jump into solving our equation, let's make sure we're all on the same page about what a quadratic equation actually is. Essentially, it's a polynomial equation of the second degree, meaning the highest power of the variable (x in our case) 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 0. These constants play a crucial role in determining the shape and position of the parabola that represents the equation when graphed. The solutions to the quadratic equation, also known as the roots or zeros, are the x-values where the parabola intersects the x-axis. These roots are of paramount importance in various mathematical and real-world applications, such as physics, engineering, and economics. Understanding the fundamental nature of quadratic equations is the first step towards mastering the techniques for solving them. Now that we have a solid grasp of what a quadratic equation is, let's move on to the specific problem we're going to solve today. Our target equation, 6x² - 30x - 84 = 0, perfectly fits the general form, and by the end of this guide, you'll be equipped with the skills to find its roots effortlessly. Remember, the beauty of mathematics lies in its systematic approach; once you understand the principles, you can apply them to solve a wide range of problems. So, stick with us as we delve deeper into the methods for solving quadratic equations and unveil the solutions to our intriguing equation.
Simplifying the Equation
The equation we're tackling is 6x² - 30x - 84 = 0. One of the smartest moves you can make when faced with a quadratic equation like this is to simplify it as much as possible before diving into the solution. Why? Because simpler equations are much easier to work with, reducing the chances of making errors and streamlining the solving process. Look closely at the coefficients – 6, -30, and -84. Notice anything special? They all share a common factor: 6! This means we can divide the entire equation by 6 without changing its solutions. This step is crucial because it not only simplifies the numbers we're dealing with but also makes the equation more manageable for the next steps. By dividing each term by 6, we transform the original equation into a much friendlier form: x² - 5x - 14 = 0. See how much cleaner that looks? This simplified form is not only easier on the eyes but also sets us up for more straightforward methods of solving, such as factoring or using the quadratic formula. Remember, simplification is a powerful tool in mathematics. It's like decluttering your workspace before starting a project – it allows you to focus on the essential parts of the problem without getting bogged down by unnecessary complexity. So, always be on the lookout for opportunities to simplify equations, whether they're quadratic or any other type. It's a habit that will serve you well in your mathematical journey. Now that we've successfully simplified our equation, we're in a prime position to explore the various methods for finding its solutions. Let's dive in and uncover the x values that make this equation true!
Method 1: Factoring
Factoring is a classic and often the quickest method for solving quadratic equations, but it works best when the equation can be factored easily. So, let's see if it works for our simplified equation, x² - 5x - 14 = 0. The idea behind factoring is to rewrite the quadratic expression as a product of two binomials. In other words, we want to find two expressions of the form (x + p) and (x + q) such that when we multiply them together, we get x² - 5x - 14. This might sound a bit like detective work, and in a way, it is! We need to find two numbers, p and q, that satisfy two conditions: their product (p * q) must equal the constant term (-14), and their sum (p + q) must equal the coefficient of the x term (-5). Let's think about the factors of -14. We have pairs like 1 and -14, -1 and 14, 2 and -7, and -2 and 7. Which of these pairs adds up to -5? Bingo! The pair 2 and -7 fits the bill perfectly. 2 multiplied by -7 gives us -14, and 2 plus -7 equals -5. This means we can rewrite our quadratic equation as (x + 2)(x - 7) = 0. Now comes the crucial step: the zero-product property. This property 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 either (x + 2) = 0 or (x - 7) = 0. Solving these two simple linear equations gives us our solutions: x = -2 and x = 7. And there you have it! We've successfully solved the quadratic equation by factoring. Factoring is a powerful tool in your mathematical arsenal, but it's not always the most efficient method for every quadratic equation. Sometimes, the factors are not immediately obvious, or the equation might not be factorable using integers. In those cases, we need to turn to other methods, such as the quadratic formula, which we'll explore next. But for now, let's celebrate our success with factoring and appreciate its elegance and efficiency when it works.
Method 2: Quadratic Formula
When factoring isn't straightforward or possible, the quadratic formula is your trusty fallback. This formula provides a guaranteed solution for any quadratic equation in the form ax² + bx + c = 0. It might look a bit intimidating at first, but once you get the hang of it, you'll see how powerful it is. The formula is: x = (-b ± √(b² - 4ac)) / (2a). Remember it, write it down, and maybe even sing it to yourself – it's that important! In our simplified equation, x² - 5x* - 14 = 0, we can identify the coefficients: a = 1, b = -5, and c = -14. Now, it's just a matter of plugging these values into the quadratic formula. Let's do it step by step. First, we calculate the discriminant, which is the part under the square root: b² - 4ac. Plugging in our values, we get (-5)² - 4 * 1 * (-14) = 25 + 56 = 81. The discriminant tells us a lot about the nature of the solutions. If it's positive (like in our case), we have two distinct real solutions. If it's zero, we have one real solution (a repeated root). And if it's negative, we have two complex solutions. Now that we have the discriminant, we can plug it back into the quadratic formula: x = (5 ± √81) / (2 * 1). The square root of 81 is 9, so we have x = (5 ± 9) / 2. This gives us two possible solutions: x = (5 + 9) / 2 = 14 / 2 = 7 and x = (5 - 9) / 2 = -4 / 2 = -2. Hey, these are the same solutions we found by factoring! This is a great confirmation that we're on the right track. The quadratic formula might seem like a longer method compared to factoring when factoring is possible, but it's a universal tool that works for any quadratic equation, even those with messy coefficients or irrational solutions. So, it's an essential technique to have in your problem-solving toolkit. With the quadratic formula under your belt, you can confidently tackle any quadratic equation that comes your way. Let's move on to another helpful technique: completing the square.
Solutions for x
Okay, guys, let's recap the journey we've taken to find the solutions for x in the equation 6x² - 30x - 84 = 0. We started by understanding what quadratic equations are and why they're so important in mathematics and beyond. We then simplified the original equation by dividing through by the common factor of 6, transforming it into the more manageable form x² - 5x - 14 = 0. This crucial step set the stage for our solution-finding adventure. Next, we explored two powerful methods for solving quadratic equations: factoring and the quadratic formula. Factoring, when applicable, is often the quickest route. We successfully factored our simplified equation into (x + 2)(x - 7) = 0, which led us to the solutions x = -2 and x = 7. But we didn't stop there! We also wielded the mighty quadratic formula, which guarantees a solution for any quadratic equation, regardless of its factorability. Plugging in our coefficients into the formula, we once again arrived at the same solutions: x = -2 and x = 7. This consistency is a testament to the robustness of both methods and reinforces our confidence in the accuracy of our results. So, what does this all mean? It means that the values x = -2 and x = 7 are the x-intercepts of the parabola represented by the equation 6x² - 30x - 84 = 0. These are the points where the parabola crosses the x-axis. They are also the roots or zeros of the quadratic equation. In practical terms, these solutions could represent various things, depending on the context of the problem. They might be the times at which a projectile hits the ground, the dimensions of a rectangle with a specific area, or the equilibrium points in a financial model. The beauty of quadratic equations lies not only in their elegant form but also in their wide range of applications. Now that we've successfully found the solutions, it's time to celebrate our mathematical prowess! You've conquered a quadratic equation, and you're one step closer to mastering algebra. Keep practicing, keep exploring, and keep enjoying the thrill of problem-solving!
Conclusion
In conclusion, guys, we've successfully navigated the world of quadratic equations and found the x values that solve 6x² - 30x - 84 = 0. We simplified the equation, employed factoring and the quadratic formula, and arrived at the solutions x = -2 and x = 7. This journey underscores the importance of understanding different solution methods and choosing the most efficient one for a given problem. Remember, math is a journey, not a destination. There's always more to learn, more to explore, and more problems to solve. So, keep practicing, keep asking questions, and keep challenging yourself. You've got this! And with that, we wrap up our exploration of solving quadratic equations. We hope you found this guide helpful and that you're now feeling more confident in your ability to tackle these types of problems. Until next time, happy solving!
