Solve X² - 10 = 30x: Solution Set Explained

Hey guys! Let's dive into a fun math problem today: finding the solution set for the equation x² - 10 = 30x. If you're scratching your head, don't worry! We're going to break it down step by step, making it super easy to understand. This isn't just about getting the answer; it's about understanding how to get there. So, buckle up, and let's get started!

Understanding Quadratic Equations

Before we jump into solving our specific equation, let's quickly recap what quadratic equations are. Quadratic equations are polynomial equations of the second degree, meaning the highest power of the variable (in our case, 'x') is 2. The standard form of a quadratic equation is ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. These equations pop up everywhere in math and real-world applications, from physics to engineering to even economics. They describe curves, trajectories, and many other phenomena.

Now, why is it essential to recognize this form? Well, it sets the stage for how we tackle these equations. We have several methods at our disposal: factoring, completing the square, and the quadratic formula. Each has its strengths and when they're most useful. Factoring is fantastic when the equation breaks down neatly. Completing the square is a bit more involved but always works. And the quadratic formula? It's our trusty universal tool, ready for any quadratic equation we throw at it. Understanding these methods and when to use them is key to mastering quadratic equations.

Consider this: a quadratic equation can have up to two real solutions, one real solution (a repeated root), or no real solutions (complex roots). This stems from the fact that we're dealing with a squared term, which can lead to multiple scenarios when finding the values of 'x' that satisfy the equation. Recognizing this possibility is crucial because it influences how we interpret our results. If we end up with a square root of a negative number, for instance, we know we're dealing with complex solutions. This understanding not only helps us solve the equations but also provides a deeper insight into the nature of mathematical solutions.

Step 1: Rearranging the Equation

Okay, so the first thing we need to do with our equation, x² - 10 = 30x, is to get it into that standard form we talked about: ax² + bx + c = 0. This makes it much easier to work with. Think of it as organizing your toolbox before starting a project – you want everything in its place, right? To do this, we need to move the 30x term from the right side of the equation to the left side. How do we do that? Simple! We subtract 30x from both sides of the equation. Remember, whatever we do to one side, we have to do to the other to keep things balanced. This is a golden rule in algebra!

So, when we subtract 30x from both sides, we get: x² - 30x - 10 = 0. Ta-da! Now our equation looks much more familiar and manageable. We've successfully rearranged it into the standard quadratic form. Here, we can clearly see that a = 1, b = -30, and c = -10. Identifying these coefficients is crucial because they're the key ingredients we'll use in the next steps, especially if we decide to use the quadratic formula. This rearrangement step might seem small, but it's a fundamental part of solving quadratic equations. It sets the stage for the methods we'll use next and ensures we're working with the equation in its most workable form. It's like laying the foundation for a building – you can't build a sturdy structure without a solid base.

Now that we've got our equation in the standard form, we're ready to start thinking about how to actually solve it. We've got a few options, and we'll explore them in the next section. But for now, pat yourselves on the back – you've completed the crucial first step! Getting the equation into the correct format is half the battle, and you've nailed it.

Step 2: Applying the Quadratic Formula

Alright, guys, now comes the exciting part – actually solving for x! We've got our equation in the standard form: x² - 30x - 10 = 0. Now, let's bring out the big guns: the quadratic formula. This formula is like the Swiss Army knife of quadratic equations; it works every time, no matter how messy the equation looks. The quadratic formula is: x = (-b ± √(b² - 4ac)) / (2a). It might look a bit intimidating at first, but trust me, it's simpler than it seems once you get the hang of plugging in the values.

Remember those coefficients we identified earlier? a = 1, b = -30, and c = -10. Now we just need to carefully substitute these values into the formula. Let's start by plugging them in: x = (-(-30) ± √((-30)² - 4 * 1 * -10)) / (2 * 1). See? We're just replacing the letters with the numbers. The next step is to simplify. First, let's deal with the negatives: -(-30) becomes +30. Then, let's calculate the discriminant, which is the part under the square root: (-30)² - 4 * 1 * -10. This simplifies to 900 + 40, which equals 940. So now our formula looks like this: x = (30 ± √940) / 2. We're getting closer!

Now, let's simplify the square root of 940. You can use a calculator for this, or you can try to simplify it by finding perfect square factors. The square root of 940 is approximately 30.66. So, we have x = (30 ± 30.66) / 2. This means we actually have two possible solutions for x, one where we add 30.66 and one where we subtract it. Let's calculate both. For the first solution, x = (30 + 30.66) / 2, which is approximately 30.33. For the second solution, x = (30 - 30.66) / 2, which is approximately -0.33. And there you have it! We've found our two solutions for x using the quadratic formula. See, it wasn't so scary after all!

Step 3: Expressing the Solution Set

Great job, guys! We've crunched the numbers and found our two solutions for x. But we're not quite done yet. The final step is to express our answer as a solution set. A solution set is simply a way of listing all the solutions to an equation in a neat and organized manner. It's like putting a bow on your hard work to show it off properly. The standard way to write a solution set is to use curly braces {} and list the solutions inside, separated by commas.

In our case, we found two solutions: approximately 30.33 and -0.33. So, our solution set looks like this: {-0.33, 30.33}. It's that simple! We've taken the individual solutions and presented them together as a set, making it clear that these are the values of x that satisfy our original equation. Sometimes, you might encounter equations with only one solution (a repeated root), in which case the solution set would contain just that one value. And sometimes, you might find equations with no real solutions (complex solutions), in which case the solution set would be an empty set, represented by {} or ∅.

Expressing the solution set is a crucial step because it provides a clear and concise answer to the problem. It's not just about finding the numbers; it's about presenting them in a way that's easy to understand. Think of it as the final polish on a piece of art – it completes the picture and makes it ready to be admired. So, whenever you solve an equation, always remember to express your answer as a solution set. It's the perfect way to wrap up your work and show off your mathematical prowess!

Alternative Methods (Factoring and Completing the Square)

While the quadratic formula is our trusty go-to, it's always good to have other tools in our toolbox. Let's briefly touch on two other methods for solving quadratic equations: factoring and completing the square. These methods aren't always the quickest route, but they can be incredibly useful in certain situations and offer a deeper understanding of quadratic equations.

Factoring is like finding the puzzle pieces that fit perfectly together. It involves breaking down the quadratic expression into two binomials that, when multiplied, give you the original equation. For example, if we had the equation x² - 5x + 6 = 0, we could factor it into (x - 2)(x - 3) = 0. This tells us that either x - 2 = 0 or x - 3 = 0, giving us the solutions x = 2 and x = 3. Factoring is fantastic when the quadratic expression breaks down neatly, but it can be tricky if the roots are irrational or complex.

Completing the square is a bit more involved but always works, just like the quadratic formula. It involves manipulating the equation to create a perfect square trinomial on one side. This method is particularly useful when the equation doesn't factor easily. The process involves taking half of the coefficient of the x term, squaring it, and adding it to both sides of the equation. This creates a perfect square trinomial, which can then be factored into a binomial squared. While completing the square can be a bit more steps, it's a powerful technique that can also be used to derive the quadratic formula itself!

Conclusion

So, guys, we've successfully navigated the world of quadratic equations and found the solution set for x² - 10 = 30x! We rearranged the equation into standard form, applied the quadratic formula, and expressed our answer as a solution set. We even touched on alternative methods like factoring and completing the square. Remember, practice makes perfect, so keep those math muscles flexed! Solving quadratic equations might seem daunting at first, but with a step-by-step approach and the right tools, you can conquer any equation that comes your way. Keep up the awesome work, and happy solving!