Solving Equations: Finding All Values Of X

Hey math enthusiasts! Today, we're diving into the world of algebra to solve the equation $-x(x+9)(x^2-9)=0$. Don't worry, it might look a little intimidating at first, but trust me, we'll break it down step by step and make it super easy to understand. Our goal here is to find all the possible values of x that make this equation true. This is a fundamental skill in mathematics, so let's get started!

Understanding the Basics: Zero Product Property

Before we jump into the equation, let's quickly review a key concept: the Zero Product Property. This property states that if the product of several factors is zero, then at least one of the factors must be zero. Think of it like this: if you multiply a bunch of numbers together and the result is zero, one or more of those numbers had to be zero. This is the cornerstone of solving equations like the one we've got.

Now, let's look at our equation: $-x(x+9)(x^2-9)=0$. Notice that we have three factors multiplied together: -x, (x+9), and (x²-9). To solve this equation, we're going to apply the Zero Product Property, meaning we'll set each of these factors equal to zero and solve for x. This will give us all the values of x that satisfy the original equation. It's like finding all the secret codes that unlock the solution! We'll begin by analyzing each factor individually. This is how we are going to determine our solutions!

Breaking Down the Equation: Step-by-Step Solutions

Alright, let's get our hands dirty and start solving! First, we have the factor -x. We set it equal to zero: -x = 0. To solve for x, we simply divide both sides by -1. This gives us x = 0. Easy peasy, right? This is our first solution, meaning if we plug 0 into the original equation, it will make the entire equation equal to zero, a true statement.

Next, we tackle the factor (x + 9). We set it equal to zero: x + 9 = 0. To solve for x, we subtract 9 from both sides, which gives us x = -9. Cool! This is our second solution. If you were to insert -9 into the initial equation, all statements would become true and equal zero. Awesome, we are making progress!

Finally, we have the factor (x² - 9). This one looks a bit different, but it's not harder. Notice that x² - 9 is a difference of squares. Remember the formula: a² - b² = (a + b)(a - b)? We can apply this here. In our case, a is x and b is 3 (since 3² = 9). So, (x² - 9) factors into (x + 3)(x - 3). Now, we have (x + 3)(x - 3) = 0. Applying the Zero Product Property again, we set each factor equal to zero.

So, from the factor x + 3, we have x + 3 = 0. Subtracting 3 from both sides gives us x = -3. And from the factor x - 3, we have x - 3 = 0. Adding 3 to both sides gives us x = 3. Woohoo! We've found our last two solutions!

The Complete Solution Set: Putting It All Together

Guys, we did it! We've successfully solved the equation $-x(x+9)(x^2-9)=0$. We found four values of x that make the equation true: 0, -9, -3, and 3. So, our solution set is { -9, -3, 0, 3 }. This means that if we substitute any of these values back into the original equation, the equation will hold true, and the result will be zero. It's like finding all the keys that unlock the door to the solution. The process involved in reaching the final solutions is easy to understand, and we have completed our objective. Great work, everyone!

To recap: We started with the equation $-x(x+9)(x^2-9)=0$. We applied the Zero Product Property, which allowed us to break down the equation into simpler factors. Then, we solved each factor individually by setting it equal to zero and solving for x. We identified a difference of squares. The important thing to remember is to always use the right property. Finally, we compiled all the solutions into a solution set. Remember, practice makes perfect. Keep working on these types of problems, and you'll become a master solver in no time!

Final Thoughts and Next Steps

So there you have it! We've successfully navigated the equation and found all possible values of x. This process highlights the importance of understanding fundamental algebraic concepts like the Zero Product Property and factoring. These skills are invaluable for tackling more complex mathematical problems. Keep in mind that as you delve deeper into mathematics, these foundations will serve you well. You'll encounter more complex equations, but the principles will remain the same. The key is to break them down into manageable steps and apply the appropriate rules and properties. Don't be afraid to practice and seek help when you need it. There are tons of resources available, from textbooks and online tutorials to study groups and teachers. The more you practice, the more confident you'll become in your ability to solve even the trickiest equations. Math is a journey, and every step you take brings you closer to mastery. Good luck, and keep exploring the amazing world of mathematics! Keep up the excellent work, and always remember to enjoy the process of learning.