Solve 17x² + 7/19x = 0: Step-by-Step Solutions

Hey guys! Today, we're diving into a fun little math problem: solving the quadratic equation 17x² + 7/19x = 0. Don't worry, it's not as intimidating as it looks! We'll break it down step-by-step so you can see exactly how it's done. Whether you're a student tackling homework or just someone who loves a good math challenge, this guide is for you. We'll use simple language and clear explanations, so you can follow along easily. So, grab your pencils and let's get started!

Understanding the Equation

Before we jump into solving, let's understand what we're dealing with. We have the equation 17x² + 7/19x = 0. This is a quadratic equation because the highest power of 'x' is 2. Quadratic equations can be a bit tricky, but they also have some cool properties that make them solvable. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants. In our case, a = 17, b = 7/19, and c = 0. Notice that 'c' is zero, which actually simplifies things quite a bit for us. When c = 0, it means we can use a neat trick called factoring to find our solutions. Factoring involves pulling out common factors from the terms in the equation. This is a crucial skill in algebra, and it's super handy for solving equations like this one. Recognizing the form of the equation helps us choose the best method to solve it. In this case, because c = 0, factoring is going to be our best friend. We're essentially looking for the values of 'x' that make the equation true, which are also known as the roots or solutions of the equation. So, with a clear understanding of our equation's structure, we're ready to move on to the next step: finding those common factors!

Step 1: Factoring Out the Common Factor

The key to solving this equation lies in factoring. Remember, factoring is like reverse distribution. We're looking for something that's common to both terms in our equation, 17x² and 7/19x. Take a close look. What do you see? That's right, both terms have 'x' in them! In fact, the lowest power of 'x' present in both terms is x to the power of 1 (which we just write as 'x'). So, 'x' is a common factor. But is that all? Well, let's think about the coefficients, 17 and 7/19. Do they have any common factors? Nope! 17 is a prime number, and 7/19 is a fraction in its simplest form. So, the only common factor we can pull out is 'x'. Now, let's actually do the factoring. We pull 'x' out of both terms: x( ). To figure out what goes inside the parentheses, we divide each term by 'x'. So, 17x² divided by x is 17x, and 7/19x divided by x is just 7/19. This gives us: x(17x + 7/19) = 0. We've successfully factored our equation! This step is crucial because it transforms our single equation into a product of two factors that equals zero. This brings us to a very important principle in mathematics: the zero-product property, which we'll use in the next step to find our solutions.

Step 2: Applying the Zero-Product Property

Okay, we've factored our equation into x(17x + 7/19) = 0. Now comes the fun part where we actually find the solutions! This is where the zero-product property comes into play. This property is super useful and says that if the product of two things is zero, then at least one of those things must be zero. Makes sense, right? If you multiply something by zero, you get zero! In our case, we have two