Finding The Positive Solution Of 0 = -x^2 + 2x + 1
Finding the solutions to quadratic equations is a fundamental skill in algebra. This article delves into solving the equation 0 = -x^2 + 2x + 1, focusing on identifying the positive solution. We will utilize the quadratic formula, a powerful tool for solving equations of the form ax^2 + bx + c = 0. This comprehensive guide will not only provide the answer but also walk you through the step-by-step process, ensuring a clear understanding of the underlying concepts. Whether you're a student tackling algebra problems or simply brushing up on your math skills, this article offers a detailed explanation and solution to this specific quadratic equation.
Understanding Quadratic Equations
Before diving into the solution, it's crucial to understand what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree, generally expressed in the form ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠0. The solutions to a quadratic equation, also known as roots or zeros, are the values of x that satisfy the equation. These solutions represent the points where the parabola defined by the quadratic equation intersects the x-axis. Quadratic equations can have two real solutions, one real solution (a repeated root), or two complex solutions. The nature of the solutions is determined by the discriminant, which is the part of the quadratic formula under the square root sign (b^2 - 4ac). If the discriminant is positive, there are two distinct real solutions; if it's zero, there is one real solution; and if it's negative, there are two complex solutions. Understanding the discriminant helps predict the type of solutions you'll encounter before even applying the quadratic formula. In real-world applications, quadratic equations model various phenomena, such as projectile motion, areas, and optimization problems. Therefore, mastering the techniques to solve them is essential for both academic and practical purposes. Recognizing the structure and properties of quadratic equations is the first step in effectively tackling problems like the one presented in this article.
The Quadratic Formula: A Powerful Tool
The quadratic formula is a cornerstone of algebra, providing a universal method for solving quadratic equations. It states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by: x = (-b ± √(b^2 - 4ac)) / 2a. This formula is derived by completing the square on the general quadratic equation and is applicable regardless of the nature of the roots (real or complex). The quadratic formula is particularly useful when factoring the quadratic expression is difficult or impossible. It ensures that you can always find the solutions, provided you correctly identify the coefficients a, b, and c. Each term in the formula plays a crucial role. The term -b accounts for the horizontal shift of the parabola, while the term √(b^2 - 4ac) determines the spread and the nature of the roots. The denominator 2a scales the solutions appropriately. Memorizing and understanding the quadratic formula is essential for any student of algebra. Its versatility and reliability make it an indispensable tool in solving quadratic equations and related problems. Moreover, familiarity with the formula provides a foundation for more advanced mathematical concepts. Applying the quadratic formula requires careful attention to detail, especially when dealing with signs and order of operations. However, with practice, it becomes a straightforward and efficient method for finding the roots of any quadratic equation.
Applying the Quadratic Formula to Our Equation
Now, let's apply the quadratic formula to the equation 0 = -x^2 + 2x + 1. First, we need to identify the coefficients a, b, and c. In this case, a = -1, b = 2, and c = 1. Plugging these values into the quadratic formula, x = (-b ± √(b^2 - 4ac)) / 2a, we get: x = (-2 ± √(2^2 - 4(-1)(1))) / 2(-1). Simplifying the expression under the square root, we have: x = (-2 ± √(4 + 4)) / -2, which further simplifies to: x = (-2 ± √8) / -2. Since √8 can be written as 2√2, the equation becomes: x = (-2 ± 2√2) / -2. Now, we can divide both terms in the numerator by -2: x = 1 ± √2. This gives us two solutions: x = 1 + √2 and x = 1 - √2. The quadratic formula has allowed us to systematically find both possible values for x that satisfy the equation. Each step, from identifying the coefficients to simplifying the final expression, is crucial for arriving at the correct solutions. In this specific problem, we have found two distinct real roots, indicating that the parabola intersects the x-axis at two points. Understanding how to substitute values into the formula and simplify the resulting expression is a fundamental skill in algebra. By mastering this process, you can confidently solve a wide range of quadratic equations.
Identifying the Positive Solution
After applying the quadratic formula, we found two solutions: x = 1 + √2 and x = 1 - √2. The problem specifically asks for the positive solution, so we need to determine which of these values is greater than zero. The square root of 2 (√2) is approximately 1.414. Therefore, 1 + √2 is approximately 1 + 1.414 = 2.414, which is clearly positive. On the other hand, 1 - √2 is approximately 1 - 1.414 = -0.414, which is negative. Thus, the positive solution is x = 1 + √2. This step highlights the importance of carefully reading the problem statement and understanding what is being asked. While the quadratic formula provides all possible solutions, the specific context of the problem may require you to select only one. In this case, identifying the positive solution involved understanding the approximate value of √2 and comparing the two solutions to zero. This type of problem-solving requires a combination of algebraic manipulation and numerical reasoning. Being able to interpret the results of a calculation and relate them back to the original question is a critical skill in mathematics. By correctly identifying the positive solution, we have successfully answered the question posed in the problem.
Final Answer and Conclusion
Therefore, the positive solution to the equation 0 = -x^2 + 2x + 1 is x = 1 + √2. This corresponds to option C in the given choices. We arrived at this answer by first understanding the structure of a quadratic equation and then applying the quadratic formula, a reliable method for finding the solutions. The quadratic formula provided us with two possible solutions, and by evaluating these solutions, we were able to identify the positive one. This problem demonstrates the importance of mastering fundamental algebraic techniques and applying them systematically. From identifying the coefficients to simplifying the expressions, each step played a crucial role in reaching the correct answer. Furthermore, it highlights the need to carefully interpret the question and select the appropriate solution based on the given criteria. Quadratic equations are a fundamental topic in algebra, and the skills learned in solving them are applicable to a wide range of mathematical and real-world problems. By understanding the underlying concepts and practicing the techniques, you can confidently tackle these types of problems. The solution x = 1 + √2 not only answers the specific question but also reinforces the broader understanding of quadratic equations and their solutions.
Final Answer: The final answer is
