Solving Quadratic Equations: Finding The Positive Solution

Hey Plastik Magazine readers! Let's dive into a classic math problem. We're going to figure out the positive solution for the equation $0 = -x^2 + 2x + 1$. This is a quadratic equation, and don't worry, it's not as scary as it sounds! We'll use the quadratic formula, a super handy tool for solving these types of equations. This is something that is used in many different aspects of life, like in engineering, physics, and even in finance to calculate investments. Understanding the quadratic formula is like having a superpower, it's pretty neat, guys.

Understanding the Quadratic Formula and the Equation

Alright, let's break this down. First off, what's a quadratic equation? It's an equation that has a variable raised to the power of 2 (like our $x^2$ here). The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. In our case, we have $a = -1$, $b = 2$, and $c = 1$. The quadratic formula is the key to unlock the solutions for x. It looks like this: x = rac{-b \pm \sqrt{b^2 - 4ac}}{2a}.

See, not so bad, right? The formula might look a bit intimidating at first glance, but once you break it down and plug in the numbers, it becomes a lot more manageable. Remember that the $\pm$ symbol means we'll get two possible answers: one where we add the square root, and one where we subtract it. This is super important because quadratic equations often have two solutions. Think of it like this: a parabola (the shape of a quadratic equation when graphed) can cross the x-axis at two different points. Now, let's plug in our numbers and get solving! This process of solving is fun since we are applying the formula to obtain the solution, and we'll see exactly how the formula works. Are you ready? Let's do it!

To make things super clear, let's explicitly identify our coefficients: $a = -1$, $b = 2$, and $c = 1$. We'll carefully substitute these values into the quadratic formula. Making sure to correctly substitute each value into its correct location in the formula is very important, this helps in the accurate calculation, so be careful. Then, we can calculate each part of the formula, starting with the part inside the square root. Following the order of operations (PEMDAS/BODMAS), we’ll simplify the expression step by step. This gives you a methodical approach that ensures you get the right solution without any silly mistakes! The next step is to calculate the square root value, and then, add and subtract the square root from the remaining parts of the formula.

This method ensures that you arrive at your answer with a surefooted understanding. Remember, the goal here is not just to get an answer, but to understand how to get the answer. This formula works every time. Once you get the hang of it, you'll be able to solve these types of equations super fast, which is pretty awesome. Also, consider the types of questions that you may be asked in the future, these can be asked in many different ways.

Step-by-Step Solution

Okay, buckle up, guys! Let's work through this step by step. First, substitute the values of $a$, $b$, and $c$ into the quadratic formula:

x = rac{-2 \pm \sqrt{2^2 - 4(-1)(1)}}{2(-1)}

Next, simplify the expression inside the square root:

x = rac{-2 \pm \sqrt{4 + 4}}{-2}

Then, simplify further:

x = rac{-2 \pm \sqrt{8}}{-2}

Now, simplify the square root. The square root of 8 can be simplified to $2\sqrt{2}$.

x = rac{-2 \pm 2\sqrt{2}}{-2}

Finally, divide both terms in the numerator by -2:

$x = 1 \mp \sqrt{2}$

This gives us two possible solutions: $x = 1 + \sqrt{2}$ and $x = 1 - \sqrt{2}$. Since the question asks for the positive solution, we know that $\sqrt{2}$ is approximately 1.414. So, $1 + \sqrt{2} \approx 2.414$ and $1 - \sqrt{2} \approx -0.414$. Therefore, the positive solution is $1 + \sqrt{2}$. The quadratic formula can look a little intimidating at first. However, once you understand how to use it, it's one of the most powerful tools in your mathematical arsenal. It’s important to understand how to manipulate expressions, substitute values correctly, and follow the order of operations to arrive at the correct result.

The steps are designed so that each part of the problem is clear and easy to follow. We carefully plugged in our values into the quadratic formula, and simplified the expression step by step to find our solution. We found two solutions, and we picked the positive one. We will always consider these steps in the future, making the process of solving these types of equations easier. Also, understanding each of these steps helps you gain a deeper understanding of the quadratic formula and how it works. That knowledge is a very useful thing.

Remember, practice makes perfect! The more you work through these problems, the more comfortable and confident you'll become. And trust me, the sense of accomplishment you get when you solve a math problem is pretty awesome.

The Correct Answer

So, the positive solution to the equation $0 = -x^2 + 2x + 1$ is $1 + \sqrt{2}$. Thus, the correct answer is C. $1 + \sqrt{2}$. Congratulations! You've successfully solved a quadratic equation using the quadratic formula! You did it, guys!

Now you know how to break down the formula and solve it. Remember that math is all about understanding the process. By following these steps and practicing more problems, you'll be able to tackle any quadratic equation that comes your way. Keep up the great work and keep exploring the amazing world of mathematics! It is like a puzzle, but a fun puzzle. Keep going, and keep growing! You guys rock!