Solving 2x²-1=0: A Step-by-Step Guide
Hey guys! Today, we're diving into the exciting world of quadratic equations, specifically tackling the equation 2x² - 1 = 0. If you've ever felt a bit puzzled about solving these types of equations, don't worry, you're in the right place! We'll break it down step by step, making sure it's super clear and easy to follow. So, grab your pencils, and let's get started!
Understanding Quadratic Equations
Before we jump into solving our specific equation, let's quickly recap what quadratic equations are all about. Quadratic equations are polynomial equations of the second degree, meaning the highest power of the variable (in our case, 'x') is 2. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and 'a' is not equal to zero. Recognizing this form is key to understanding how to solve these equations.
In our equation, 2x² - 1 = 0, we can see that 'a' is 2, 'b' is 0 (since there's no 'x' term), and 'c' is -1. Identifying these coefficients is a crucial first step. Now, why are we even bothering with these equations? Well, quadratic equations pop up in various real-world scenarios, from physics problems involving projectile motion to engineering designs and even financial calculations. So, mastering them is super valuable.
There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. For our equation, we'll use a straightforward algebraic approach, but it's good to know there are multiple tools in our toolbox. Remember, the goal is 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 dive into solving 2x² - 1 = 0 and uncover those roots!
Isolating the x² Term
The first step in solving the equation 2x² - 1 = 0 is to isolate the x² term. Think of it like peeling an onion – we're removing the outer layers to get to the core. To do this, we'll perform a couple of simple algebraic manipulations. Our goal is to get the x² term all by itself on one side of the equation.
First, we need to get rid of the -1. How do we do that? We add 1 to both sides of the equation. Remember, whatever we do to one side, we must do to the other to keep the equation balanced. This gives us: 2x² - 1 + 1 = 0 + 1, which simplifies to 2x² = 1. Great! We've made progress.
Now, we have 2x² = 1. We still need to isolate x², so we need to get rid of the 2 that's multiplying it. To do this, we'll divide both sides of the equation by 2. This gives us: (2x²)/2 = 1/2, which simplifies to x² = 1/2. Awesome! We've successfully isolated the x² term. This is a major milestone in solving our equation.
By isolating x², we've set ourselves up for the next crucial step: finding the square root. This step will reveal the values of 'x' that satisfy our equation. So, we're well on our way to cracking this quadratic equation. Stick with me, and let's move on to the next step!
Finding the Square Root
Now that we've successfully isolated x² in the equation x² = 1/2, it's time to find the square root. This is the step where we'll uncover the actual values of 'x'. Remember, finding the square root is the inverse operation of squaring a number. So, if x² is equal to 1/2, then x is equal to the square root of 1/2.
But here's a crucial point: when we take the square root, we need to consider both the positive and negative roots. Why? Because both a positive number and its negative counterpart, when squared, will give a positive result. For example, both 2² and (-2)² equal 4. So, when finding the square root, we need to account for both possibilities.
So, we have x = ±√(1/2). This means x can be either the positive square root of 1/2 or the negative square root of 1/2. Let's break this down a bit further. The square root of 1/2 can be written as √1 / √2. Since √1 is simply 1, we have 1 / √2. However, it's generally considered good practice to rationalize the denominator, which means getting rid of the square root in the denominator.
To rationalize the denominator, we multiply both the numerator and the denominator by √2. This gives us (1 * √2) / (√2 * √2), which simplifies to √2 / 2. So, the square root of 1/2 is √2 / 2. Remembering to consider both positive and negative roots, our solutions for x are x = √2 / 2 and x = -√2 / 2. We're almost there – let's summarize our findings!
Summarizing the Solutions
Alright, guys, we've reached the finish line! We've successfully navigated the steps to solve the equation 2x² - 1 = 0. Let's take a moment to recap what we've done and clearly state our solutions. This is a critical step because it ensures we understand the entire process and can confidently apply it to other quadratic equations.
We started with the equation 2x² - 1 = 0. Our first move was to isolate the x² term. We added 1 to both sides, giving us 2x² = 1. Then, we divided both sides by 2, resulting in x² = 1/2. So far, so good!
Next, we found the square root of both sides of the equation. This is where we remembered the important point about considering both positive and negative roots. We found that x = ±√(1/2). We then simplified √(1/2) to √2 / 2 by rationalizing the denominator.
Therefore, our solutions are x = √2 / 2 and x = -√2 / 2. These are the two values of 'x' that make the equation 2x² - 1 = 0 true. We can write this more concisely as x = ±√2 / 2. Congratulations! You've solved a quadratic equation.
By following these steps, you can tackle a variety of similar equations. Remember, the key is to isolate the x² term, find the square root (considering both positive and negative roots), and simplify your answer. Now you've got another tool in your math belt. Keep practicing, and you'll become a quadratic equation-solving pro in no time!
