Solving N² - 21n = -98: Find All Real Answers
Hey there, math enthusiasts and problem-solvers! Ever found yourself staring at an equation like n² - 21n = -98 and wondering, "How on earth do I tackle this beast to find all its real solutions?" Well, you're in luck! Today, we're diving deep into the fascinating world of quadratic equations, specifically focusing on how to confidently solve n² - 21n = -98 and uncover its hidden real solutions. This isn't just about crunching numbers; it's about understanding the logic, choosing the right tools, and feeling empowered to conquer any quadratic that comes your way. Whether you're a student looking to ace your next exam or just someone curious about the elegant dance of numbers, this article is crafted just for you. We'll break down the steps, use a friendly, conversational tone, and ensure you walk away with a solid grasp of how to find those elusive real solutions. So, grab your favorite beverage, get comfy, and let's embark on this mathematical adventure together. Trust me, by the end of this, solving n² - 21n = -98 will feel like a walk in the park. We'll go through the entire process, from setting up the equation correctly to applying the powerful quadratic formula, and even show you how to verify your answers. Ready to become a quadratic equation master? Let's do this!
Understanding Quadratic Equations: A Quick Refresher for Finding Real Solutions
Before we jump into solving n² - 21n = -98, let's first get cozy with what a quadratic equation actually is. Think of it as a special kind of algebraic equation where the highest power of the variable (in our case, 'n') is 2. The most common way you'll see a quadratic equation is in its standard form: ax² + bx + c = 0, where 'a', 'b', and 'c' are coefficients (just numbers!) and 'a' can't be zero. If 'a' were zero, it wouldn't be quadratic anymore, right? It would just be a linear equation, and that's a whole other ball game. Why are these equations so important? Well, guys, quadratic equations pop up everywhere in the real world – from figuring out the trajectory of a thrown ball (hello, physics!), to designing efficient parabolic satellite dishes, or even optimizing profits in business models. They're fundamental! Knowing how to find all real solutions for these equations is a core skill that opens doors to understanding so many real-life phenomena. When we talk about "real solutions," we're simply looking for values of 'n' that are actual numbers you can find on a number line, not those cool but sometimes tricky imaginary numbers. Our goal with n² - 21n = -98 is to identify these real number values that make the equation true. Getting comfortable with the standard form is the very first, and arguably most crucial, step in preparing any quadratic equation for solution. It's like preparing your ingredients before you start cooking – you want everything neatly organized and ready to go. So, let's keep that standard form, ax² + bx + c = 0, firmly in mind as we move forward. It's our guiding star for solving n² - 21n = -98 and similar problems. Understanding this foundation will make the rest of our journey much smoother, ensuring we can confidently find all real solutions without breaking a sweat.
Setting Up Our Equation for Success: Transforming n² - 21n = -98
Alright, team, the next critical step in solving n² - 21n = -98 is getting it into that beautiful standard form: ax² + bx + c = 0. Right now, our equation looks a little different: -21n + n² = -98. Notice how the terms aren't neatly arranged, and one side isn't zero. No worries, though! This is a super easy fix. The main goal here is to gather all the terms on one side of the equation, leaving a clean zero on the other side. Think of it as tidying up your workspace before you start a big project. A cluttered workspace leads to cluttered thinking! So, to start, let's rearrange the terms on the left side to put the n² term first, which is standard practice and just looks cleaner: n² - 21n = -98. See, already looking better! Now, for that all-important zero on the right side. To achieve this, we need to move the -98 from the right side to the left side. How do we do that? By performing the opposite operation. Since it's currently subtracting 98 (or rather, is a negative 98), we'll add 98 to both sides of the equation. Remember, whatever you do to one side of an equation, you must do to the other to keep it balanced. This fundamental rule is key to solving n² - 21n = -98 without messing things up. So, if we add 98 to both sides, our equation transforms into: n² - 21n + 98 = 0. Boom! There it is! Our equation is now perfectly in the standard ax² + bx + c = 0 form. From this, we can easily identify our coefficients: a = 1 (because there's an implied '1' in front of n²), b = -21 (don't forget that minus sign!), and c = 98. Having these values clearly identified is absolutely crucial for the next step, where we'll unleash the power of the quadratic formula to find all real solutions. This setup process, though seemingly simple, lays the entire foundation for a successful solution. Don't ever skip or rush this part, guys, because getting it right here means smooth sailing ahead for solving n² - 21n = -98!
The Power of the Quadratic Formula: Our Go-To Method for Solving n² - 21n = -98
Alright, math warriors, we've prepped our equation, and now it's time for the main event: applying the incredible quadratic formula to solve n² - 21n + 98 = 0 and find all real solutions. This formula is like a superhero tool for quadratics; it will always give you the solutions, no matter how tricky the numbers are. You might remember it, or maybe you're seeing it for the first time, but either way, commit it to memory: n = [-b ± √(b² - 4ac)] / 2a. Looks a bit intimidating, right? Don't sweat it! We're going to break it down piece by piece. First things first, let's recall the coefficients we identified from our perfectly set-up equation, n² - 21n + 98 = 0: we have a = 1, b = -21, and c = 98. See how important that setup step was? Now, we just plug these values directly into our quadratic formula. It's like following a recipe! Before we calculate the whole thing, let's focus on the part under the square root, called the discriminant (b² - 4ac). The discriminant is super important because it tells us what kind of solutions we're going to get. If it's positive, we get two real solutions (exactly what we're looking for!). If it's zero, we get one real solution (or two identical ones). If it's negative, we get two complex solutions (which aren't real, but still valid in the complex number system). So, let's calculate our discriminant for n² - 21n + 98 = 0: (-21)² - 4(1)(98). That's 441 - 392. And guess what? 441 - 392 = 49. Since 49 is a positive number, we know we're definitely going to find two real solutions for 'n' – excellent news! Now that we have the discriminant, we can continue plugging everything into the full formula. Remember, n = [ -b ± √(b² - 4ac) ] / 2a. Substituting our values: n = [ -(-21) ± √(49) ] / (2 * 1). Simplify that, and you get: n = [ 21 ± 7 ] / 2. This is where the magic happens and our two solutions emerge! The '±' symbol means we have two possibilities: one where we add 7 and one where we subtract 7. This is how we extract both of our real solutions for 'n' when solving n² - 21n = -98. Keep going, you're almost there!
Step-by-Step Calculation: Applying the Formula to Find Our Real Solutions
Okay, guys, we're on the home stretch of solving n² - 21n = -98! We've got our simplified expression from the quadratic formula: n = [ 21 ± 7 ] / 2. Now, let's break this down into our two distinct real solutions. This is the moment we've been waiting for! The first solution, let's call it n₁, comes from taking the plus sign:
- n₁ = (21 + 7) / 2
- n₁ = 28 / 2
- n₁ = 14
Bingo! There's our first real solution: n = 14. Pretty straightforward, right? Now, for the second real solution, let's call it n₂, we take the minus sign:
- n₂ = (21 - 7) / 2
- n₂ = 14 / 2
- n₂ = 7
And voilà! Our second real solution is n = 7. So, the real solutions for the equation n² - 21n = -98 are n = 14 and n = 7. See? It wasn't nearly as scary as it might have looked initially! The quadratic formula really does all the heavy lifting once you've correctly identified your 'a', 'b', and 'c' values and performed the substitution carefully. This systematic approach ensures that you can reliably find all real solutions for any quadratic equation, not just this specific one. Always remember the power of patience and precision when working through these calculations. A tiny mistake in a sign or a calculation can lead you down the wrong path, so double-checking your work, especially when dealing with the discriminant and the final division, is always a super smart move. You've successfully applied one of the most fundamental tools in algebra, demonstrating your capability to solve n² - 21n = -98 completely. Give yourself a pat on the back!
Verifying Our Solutions and What They Mean
Finding the solutions is awesome, but a true math pro always verifies their work. This step is super important, guys, because it gives you confidence in your answers and helps catch any silly mistakes. To verify our solutions for n² - 21n = -98, we simply plug each of our found values back into the original equation and see if it holds true. Our original equation was -21n + n² = -98. Let's start with our first solution, n = 14:
- Substitute n = 14: -21(14) + (14)² = -98
- Calculate: -294 + 196 = -98
- Check: -98 = -98
Perfect! The equation holds true for n = 14, so we know that's a correct real solution. See? That's the power of verification! Now, let's do the same for our second solution, n = 7:
- Substitute n = 7: -21(7) + (7)² = -98
- Calculate: -147 + 49 = -98
- Check: -98 = -98
Fantastic! Both solutions, n = 14 and n = 7, correctly satisfy the original equation. This means we've successfully solved n² - 21n = -98 and confidently found all real solutions. But what does