Solve X² + 2x = 6 Using The Quadratic Formula
Hey there, math enthusiasts! Let's dive into a classic quadratic equation problem and break it down together. We're going to use the quadratic formula, a trusty tool in any mathematician's arsenal, to solve for x in the equation x² + 2x = 6. Don't worry if it looks intimidating at first; we'll take it one step at a time. Our goal is to not only find the correct answer but also to understand why it's the correct answer. So, grab your pencils, and let's get started!
Understanding the Quadratic Equation
Before we jump into the formula itself, let's make sure we're all on the same page about what a quadratic equation is. In essence, a quadratic equation is a polynomial equation of the second degree. This means that 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. It's super important to get the equation into this standard form because the quadratic formula is specifically designed to work with equations formatted this way. Identifying a, b, and c correctly is the first crucial step to solving the equation. These coefficients are the keys that unlock the solution, and messing them up can lead to a completely wrong answer. Trust me, I've been there! So, double-check those values before moving on.
In our specific equation, x² + 2x = 6, we first need to rewrite it in the standard form. To do this, we subtract 6 from both sides, giving us x² + 2x - 6 = 0. Now, we can clearly see that a = 1 (the coefficient of x²), b = 2 (the coefficient of x), and c = -6 (the constant term). Notice that c is negative here, which is a common pitfall if you're not careful with the signs. Remember, the sign is just as important as the number itself! With these values in hand, we're ready to tackle the quadratic formula. Think of these coefficients as the ingredients in a recipe; you need them in the right amounts to get the desired result. So, take a moment to really make sure you've got them right. It's like making sure you have the right spices before you start cooking – it can make all the difference in the final dish!
The Quadratic Formula: Our Trusty Tool
Now for the star of the show: the quadratic formula. This formula is a powerful tool that gives us the solutions (also called roots) for any quadratic equation in the standard form. It looks a bit intimidating at first glance, but trust me, it's your best friend when dealing with quadratics. The formula is: x = (-b ± √(b² - 4ac)) / 2a. See? Not so scary once you break it down! The ± symbol means we'll actually get two solutions, one with addition and one with subtraction. This is because quadratic equations can have up to two real roots. Think of the square root part as the engine of the formula; it's where all the action happens. And the 2a in the denominator? That's the final touch that brings everything together. The beauty of this formula is that it works every single time, no matter how messy the equation looks. It's like having a universal key that unlocks any quadratic puzzle. But, like any tool, it's important to know how to use it correctly. That's where we come in! We'll guide you through the process step-by-step, making sure you understand each part of the formula and how it applies to our specific equation.
The part under the square root, b² - 4ac, is called the discriminant. This little expression tells us a lot about the nature of the solutions. If the discriminant is positive, we have two distinct real roots. If it's zero, we have exactly one real root (a repeated root). And if it's negative, we have two complex roots. But don't worry about complex roots for now; we're focusing on real solutions in this problem. Understanding the discriminant is like having a weather forecast for the solutions. It gives you a heads-up about what to expect. So, before you even plug the numbers into the full formula, you can get a sense of what kind of solutions you're going to find. Pretty neat, huh? The quadratic formula might seem like just a bunch of symbols and numbers, but it's actually a powerful way to understand the behavior of quadratic equations. And once you've mastered it, you'll feel like a true math wizard!
Applying the Formula: Step-by-Step
Alright, let's put the quadratic formula to work! We've already identified that a = 1, b = 2, and c = -6. Now, we'll carefully substitute these values into the formula. This is where precision is key! It's like following a recipe – you need to add the ingredients in the right order and in the right amounts. A tiny mistake here can throw off the whole solution. So, double-check each substitution as you go. First, we replace b with 2, a with 1, and c with -6 in the formula: x = (-2 ± √(2² - 4 * 1 * -6)) / (2 * 1). Notice how we've replaced each variable with its corresponding value. The parentheses are super important here, especially when dealing with negative numbers. They help us keep track of the signs and avoid silly mistakes. Next, we simplify the expression inside the square root. 2² is 4, and 4 * 1 * -6 is -24. So, we have 4 - (-24), which is the same as 4 + 24, giving us 28. Now our equation looks like this: x = (-2 ± √28) / 2. We're getting closer! The square root part is where things often get a little tricky, but we'll handle it like pros. We've simplified the discriminant and now it's time to move to the next steps. Remember, patience and attention to detail are your best friends in this process. Don't rush, and don't be afraid to double-check your work. It's much better to take a little extra time and get it right than to rush and make a mistake.
Now, we need to simplify the square root of 28. We can do this by finding the prime factorization of 28, which is 2 * 2 * 7, or 2² * 7. This means that √28 is the same as √(2² * 7), which can be simplified to 2√7. Remember your rules of radicals! This step is like unwrapping a present – we're revealing the hidden structure within the square root. So, our equation now looks like this: x = (-2 ± 2√7) / 2. We're almost there! The final step is to simplify the entire expression. Notice that both terms in the numerator have a factor of 2, and so does the denominator. We can divide each term by 2 to simplify the fraction. This is like putting the finishing touches on a masterpiece. We've done all the hard work, and now we're just polishing it up to make it shine. Dividing -2 by 2 gives us -1, and dividing 2√7 by 2 gives us √7. So, our final solutions are x = -1 ± √7. We did it! We've successfully navigated the quadratic formula and found the solutions to our equation. Take a moment to celebrate your hard work – you've earned it! And remember, the more you practice, the easier this process will become. You'll be solving quadratic equations in your sleep in no time.
Identifying the Correct Answer
Looking at the options provided, we can see that D. x = -1 ± √7 matches our solution perfectly. Hooray! We've not only solved the equation but also identified the correct answer from the given choices. This is a crucial step in any math problem – making sure your answer actually lines up with the options available. It's like having a map and making sure you've reached the right destination. Double-checking your answer can save you from making careless mistakes and missing out on points. So, always take that extra moment to compare your solution with the choices provided. But more importantly, let's talk about why the other options are incorrect. This is just as valuable as finding the right answer because it helps us understand the common mistakes people make and how to avoid them. Option A, x = -1 ± √7/7, has a fraction under the square root, which we didn't encounter in our solution. This suggests a mistake in simplifying the square root or in the final division step. Option B, x = 1 ± √7/7, has the wrong sign for the constant term. Remember, we ended up with -1, not 1. This could be due to a sign error when substituting into the quadratic formula or during the simplification process. And option C, x = 1 ± √7, also has the wrong sign for the constant term. By understanding why these options are incorrect, we're reinforcing our understanding of the correct method. It's like learning from your mistakes before you even make them! So, let's give ourselves a pat on the back for not only finding the right answer but also understanding the entire process.
Key Takeaways and Practice Tips
So, what have we learned today? We've tackled a quadratic equation using the quadratic formula, and we've emerged victorious! We've seen how important it is to correctly identify a, b, and c, to carefully substitute these values into the formula, and to simplify the resulting expression step-by-step. We've also learned the importance of double-checking our work and comparing our solution with the given options. But the real key to mastering the quadratic formula is practice, practice, practice! The more you use it, the more comfortable you'll become with it. It's like learning a new language – the more you speak it, the more fluent you'll become. So, don't be afraid to tackle more quadratic equations. Look for practice problems online, in textbooks, or even make up your own! Try changing the coefficients and see how the solutions change. Experiment with different types of quadratic equations, some with real solutions, some with no real solutions, and some with repeated solutions. The more you explore, the deeper your understanding will become. And remember, math is not just about finding the right answer; it's about the journey of discovery. It's about the thrill of solving a puzzle, the satisfaction of mastering a new skill, and the confidence that comes from knowing you can tackle any challenge that comes your way. So, keep practicing, keep exploring, and keep having fun with math!
Practice Problems
Here are a few practice problems to get you started:
- Solve 2x² - 5x + 2 = 0
- Find the solutions for x² + 4x + 4 = 0
- Use the quadratic formula to solve 3x² + 2x - 1 = 0
Go ahead and give these a try. Remember, the key is to take it step by step, and don't be afraid to make mistakes. That's how we learn! And if you get stuck, don't worry. Go back and review the steps we've covered in this article. Or, reach out to a friend, a teacher, or an online community for help. There are plenty of resources available to support you on your math journey. And most importantly, remember to celebrate your successes along the way. Every problem you solve is a step forward, and every step brings you closer to mastering the quadratic formula and other mathematical concepts. So, keep up the great work, and remember, you've got this!
