Hey there, math enthusiasts! Today, we're diving deep into the fascinating world of quadratic equations. Specifically, we're going to tackle the equation 2x² + 3x = 1. Don't worry if it looks a bit intimidating at first glance; we'll break it down step by step and find those exact solutions. So, grab your calculators (or your mental math muscles) and let's get started!
Understanding Quadratic Equations
Before we jump into solving, let's quickly recap what a quadratic equation actually is. In essence, a quadratic equation is a polynomial equation of the second degree. This means 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 (otherwise, it would be a linear equation).
Now, why are we so interested in these equations? Well, quadratic equations pop up everywhere in the real world, from physics and engineering to finance and computer science. They can model the trajectory of a projectile, the shape of a satellite dish, or even the growth of a population. Understanding how to solve them opens doors to understanding and solving a myriad of real-world problems. There are several methods we can use to solve quadratic equations, including factoring, completing the square, and the quadratic formula. Each method has its strengths and weaknesses, and the best approach often depends on the specific equation we're dealing with. For instance, factoring is a quick and elegant solution when the equation can be easily factored, but it's not always applicable. Completing the square is a more general method that works for any quadratic equation, but it can be a bit more involved. And then there's the quadratic formula, a powerful tool that guarantees a solution, no matter how messy the equation looks. In this article, we'll primarily focus on using the quadratic formula to solve our equation, as it's a reliable and universally applicable method. However, we might also touch upon other methods to give you a broader understanding of how these equations can be tackled. So, stay tuned as we delve deeper into the world of quadratic equations and uncover the secrets to solving them!
Setting Up the Equation
The first thing we need to do is get our equation, 2x² + 3x = 1, into the standard form (ax² + bx + c = 0). This is crucial because the standard form allows us to easily identify the coefficients (a, b, and c) that we'll need for our solution methods, especially the quadratic formula. To achieve this, we simply subtract 1 from both sides of the equation. This gives us:
2x² + 3x - 1 = 0
Great! Now we can clearly see that:
- a = 2
- b = 3
- c = -1
These values are the keys we need to unlock the solutions to our equation. With the equation in standard form and the coefficients identified, we're perfectly poised to apply the quadratic formula. This formula, a cornerstone of algebra, provides a direct pathway to the solutions of any quadratic equation. Before we jump into the formula itself, it's worth appreciating why this step of rearranging the equation is so important. Putting the equation in standard form not only makes it easier to identify the coefficients but also ensures that we're using the correct values in the subsequent steps. A slight misidentification of a coefficient can lead to completely different solutions, so this meticulous preparation is key to accuracy. Moreover, understanding this rearrangement process reinforces the fundamental principle of maintaining equality in equations. By performing the same operation (in this case, subtracting 1) on both sides, we ensure that the equation remains balanced and that the solutions we find are valid. So, as we move forward, remember the importance of this initial step. It's a small but significant detail that sets the stage for a successful solution.
The Mighty Quadratic Formula
Okay, guys, now for the star of the show: the quadratic formula! This formula is your best friend when it comes to solving quadratic equations, especially those that don't factor easily. It's a bit of a mouthful, but trust me, it's worth memorizing:
x = (-b ± √(b² - 4ac)) / 2a
See? Not so scary, right? Now, let's break it down. We've already identified our 'a', 'b', and 'c' values from the previous step. All we need to do is plug them into the formula and simplify. The ± symbol means we'll actually get two solutions: one where we add the square root part and one where we subtract it. This is because quadratic equations can have up to two real solutions.
The quadratic formula isn't just a magic trick; it's derived from the method of completing the square, a technique for rewriting quadratic equations in a form that's easy to solve. The formula essentially encapsulates the steps of completing the square, providing a direct route to the solutions. The expression inside the square root, b² - 4ac, is particularly important. It's called the discriminant, and it tells us a lot about the nature of the solutions. If the discriminant is positive, we have two distinct real solutions. If it's zero, we have exactly one real solution (a repeated root). And if it's negative, we have two complex solutions (which involve imaginary numbers). In our case, we'll see that the discriminant is positive, indicating that we'll find two different real solutions for our equation. As we move forward, we'll carefully substitute the values of 'a', 'b', and 'c' into the formula and simplify the expression step by step. This process will not only lead us to the solutions but also reinforce our understanding of how the formula works. So, let's dive in and see the quadratic formula in action!
Plugging in the Values
Time to get our hands dirty! We'll substitute our values (a = 2, b = 3, c = -1) into the quadratic formula:
x = (-3 ± √(3² - 4 * 2 * -1)) / (2 * 2)
Now, let's simplify this step by step. First, we'll tackle the expression inside the square root:
3² - 4 * 2 * -1 = 9 + 8 = 17
So, our equation now looks like this:
x = (-3 ± √17) / 4
This step-by-step substitution is crucial for avoiding errors. By carefully plugging in the values and simplifying each part of the equation, we ensure that we're on the right track. It's like building a house – each brick needs to be laid correctly for the structure to be sound. In this case, each numerical substitution and arithmetic operation is a brick in our solution. The order of operations is also paramount here. We first deal with the exponent (3²), then the multiplication (-4 * 2 * -1), and finally the addition (9 + 8). Following this order ensures that we arrive at the correct value for the discriminant. The square root of 17 is an irrational number, meaning it cannot be expressed as a simple fraction. This tells us that our solutions will also be irrational, which is perfectly fine. It simply means that the solutions will have a non-repeating, non-terminating decimal representation. As we continue to simplify, we'll keep the square root of 17 as it is, as it's the most accurate way to represent this number. We're now on the verge of finding our two solutions, and the careful work we've done so far will pay off as we move to the final steps.
Finding the Two Solutions
Remember that ± symbol? It means we have two solutions to find. Let's calculate them separately:
Solution 1 (using the + sign):
x₁ = (-3 + √17) / 4
Solution 2 (using the - sign):
x₂ = (-3 - √17) / 4
These are the exact solutions to our equation! We can leave them in this form, or if we need decimal approximations, we can use a calculator. It's important to note that these solutions are irrational numbers, meaning they cannot be expressed as a simple fraction. They have decimal representations that go on forever without repeating. Leaving the solutions in their exact form (with the square root) is often preferred in mathematics, as it avoids any rounding errors and preserves the precise values. However, in practical applications, decimal approximations might be more useful. For example, if we were using these solutions to determine the dimensions of a physical object, we would need to work with decimal values. The fact that we have two distinct real solutions aligns with our earlier observation about the discriminant being positive. This consistency between the discriminant and the nature of the solutions is a powerful check on our work. It's like having a built-in error detector that alerts us if something goes wrong along the way. As we conclude this step, we've successfully found the exact solutions to the quadratic equation. But our journey doesn't end here. It's always a good practice to verify our solutions to ensure their correctness.
Verifying the Solutions
To make sure we haven't made any mistakes, let's verify our solutions. We'll plug each one back into the original equation (2x² + 3x = 1) and see if it holds true. This is a crucial step in any mathematical problem-solving process. It's like double-checking your work before submitting a final report. Plugging the solutions back into the original equation allows us to catch any potential errors in our calculations. If the equation doesn't hold true, it means we've made a mistake somewhere along the way, and we need to go back and review our steps. For irrational solutions like the ones we've found, verifying can be a bit tricky without a calculator. We would need to square the expressions, multiply them by 2, add 3 times the expression, and see if the result is close to 1. Due to the nature of irrational numbers, we might not get exactly 1, but we should get a value very close to it. Alternatively, we can use a calculator to get decimal approximations of our solutions and plug those into the original equation. This will give us a numerical check on our work. If the decimal approximations satisfy the equation, we can be confident that our exact solutions are correct. Verification isn't just about finding errors; it's also about building confidence in our work. By confirming that our solutions are valid, we reinforce our understanding of the problem-solving process and the concepts involved. It's a valuable habit to develop, and it will serve you well in all your mathematical endeavors. So, let's take a moment to appreciate the importance of verification as we wrap up this exploration of quadratic equations.
Conclusion
And there you have it! We've successfully navigated the world of quadratic equations and found the exact solutions to 2x² + 3x = 1. We learned about the standard form of a quadratic equation, the power of the quadratic formula, and the importance of verifying our results. Quadratic equations might seem daunting at first, but with a systematic approach and the right tools, they become much more manageable. Remember, practice makes perfect, so keep exploring and solving! You've now equipped yourself with a valuable tool for tackling a wide range of mathematical problems. The journey through this equation has highlighted several key concepts in algebra, including the structure of quadratic equations, the power of formulas, and the importance of careful calculation. Each step we took, from rearranging the equation to verifying the solutions, reinforced these concepts and built a deeper understanding of the underlying principles. As you continue your mathematical journey, remember that problem-solving is not just about finding the answer; it's about the process of learning and growing. Embrace the challenges, ask questions, and never stop exploring. The world of mathematics is full of fascinating discoveries, and you're well on your way to uncovering them. So, keep practicing, keep learning, and keep solving!