Solving 3x² + 12x = 0 A Step-by-Step Guide

Hey guys! Today, we're going to break down how to solve the quadratic equation 3x² + 12x = 0. Don't worry, it's not as scary as it looks! We'll go through it step-by-step so you can totally nail this. Quadratic equations might seem intimidating, but once you understand the basic principles and techniques, you'll be solving them like a pro. This particular equation is a great example because it demonstrates a common type of quadratic equation that can be solved relatively easily using factoring. So, let's dive in and get started!

Understanding Quadratic Equations

Before we jump into solving, let's quickly recap what a quadratic equation actually is. A quadratic equation is basically a polynomial equation of the second degree. What does that mean? Well, it means the highest power of the variable (usually x) in the equation is 2. The standard form of a quadratic equation is written as ax² + bx + c = 0, where a, b, and c are constants (numbers), and 'a' cannot be zero (otherwise, it wouldn't be a quadratic equation anymore!). Now, in our equation, 3x² + 12x = 0, we can see that 'a' is 3, 'b' is 12, and 'c' is 0. Understanding this form is crucial because it helps us identify the coefficients that we'll use in our solution methods. There are several ways to solve quadratic equations, such as factoring, completing the square, and using the quadratic formula. Each method has its strengths and is suitable for different types of equations. For instance, factoring is often the quickest method when the equation can be easily factored, which is the case for our example today. Recognizing the structure of a quadratic equation is the first step toward mastering its solution. Remember, the goal is always to find the values of x that make the equation true. These values are also known as the roots or solutions of the equation. So, let's move on to the fun part – solving!

Step 1: Factoring Out the Common Factor

The key to solving 3x² + 12x = 0 is factoring. Factoring is like the superhero move of solving quadratic equations when it works. In this case, it totally does! Look closely at the equation: 3x² + 12x = 0. Can you see a common factor in both terms? Yup, you guessed it – both terms have a common factor of 3x. So, let's factor that out. When we factor out 3x, we're essentially dividing both terms by 3x and writing the equation in a new form. 3x² divided by 3x is x, and 12x divided by 3x is 4. Therefore, we can rewrite the equation as 3x(x + 4) = 0. This is a crucial step because it transforms our quadratic equation into a product of two factors that equals zero. Why is this important? Because of 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 mathematical terms, if A * B = 0, then either A = 0 or B = 0 (or both). This property is the foundation for solving many factored equations, and it's what we'll use next to find our solutions for x. Factoring simplifies the equation and makes it much easier to handle, turning a potentially complex problem into a straightforward one. So, by identifying and factoring out the common factor, we've made significant progress towards finding the values of x that satisfy our equation.

Step 2: Applying the Zero-Product Property

Okay, now we've got 3x(x + 4) = 0. Remember the zero-product property we talked about? This is where it comes into play! This property tells us that if the product of two things is zero, then at least one of those things must be zero. In our case, the "two things" are 3x and (x + 4). So, either 3x = 0 or (x + 4) = 0. This is a huge breakthrough because it splits our one quadratic equation into two simpler, linear equations. Now, let's tackle each of these equations separately. First, we have 3x = 0. To solve for x, we simply divide both sides of the equation by 3. This gives us x = 0 / 3, which simplifies to x = 0. So, that's our first solution! Next, we have (x + 4) = 0. To solve for x here, we subtract 4 from both sides of the equation. This gives us x = -4. And that's our second solution! By applying the zero-product property, we've managed to break down the problem into manageable parts and find two possible values for x that make the original equation true. This is a common technique in algebra and a powerful tool for solving equations. So, we're almost there – just one more step to wrap things up and make sure we've got the right answers.

Step 3: Finding the Solutions

So, after applying the zero-product property, we found two potential solutions: x = 0 and x = -4. But we're not done just yet! It's always a good idea to check our answers to make sure they actually work. This is especially important in more complex equations, but it's a great habit to get into. Let's start by plugging x = 0 back into the original equation: 3x² + 12x = 0. Substituting x with 0, we get 3(0)² + 12(0) = 0. Simplifying, we have 3(0) + 0 = 0, which becomes 0 + 0 = 0. And guess what? It's true! So, x = 0 is definitely a solution. Now, let's try x = -4. Plugging this into the original equation, we get 3(-4)² + 12(-4) = 0. Let's break this down: (-4)² is 16, so we have 3(16) + 12(-4) = 0. This simplifies to 48 - 48 = 0, which is also true! So, x = -4 is also a solution. Awesome! We've checked both solutions, and they both satisfy the original equation. This confirms that our solutions are correct. Therefore, the solutions to the quadratic equation 3x² + 12x = 0 are x = 0 and x = -4. By checking our answers, we've not only ensured accuracy but also gained confidence in our solution process. And that's it – we've successfully solved the quadratic equation! You did it!

Conclusion

Alright, we did it! Solving the quadratic equation 3x² + 12x = 0 was a breeze once we broke it down into steps. First, we understood the basic form of a quadratic equation and identified the coefficients. Then, we used the super handy factoring method to simplify the equation. We factored out the common factor of 3x, which gave us 3x(x + 4) = 0. Next, we applied the zero-product property, which allowed us to split the equation into two simpler equations: 3x = 0 and x + 4 = 0. Solving these, we found potential solutions of x = 0 and x = -4. Finally, we checked our answers by plugging them back into the original equation to make sure they worked. And they did! So, we confidently concluded that the solutions are x = 0 and x = -4. Remember, practice makes perfect! The more you solve these types of equations, the easier they'll become. Factoring is a powerful tool in algebra, and understanding how to use it will help you tackle all sorts of problems. Keep practicing, and you'll become a quadratic equation-solving wizard in no time! This step-by-step guide should give you a solid foundation for tackling similar problems. If you ever get stuck, just remember these steps: identify the equation type, look for common factors, apply the zero-product property, and always check your answers. You've got this!