Solving 2x^2-5x+1=3 A Step-by-Step Guide

#h1 Solve for x in the equation 2x^2-5x+1=3: A Comprehensive Guide

In the realm of mathematics, solving quadratic equations is a fundamental skill. This article delves into the process of finding the solutions for x in the equation 2x² - 5x + 1 = 3. We will explore the steps involved, the underlying principles, and the application of the quadratic formula to arrive at the correct solutions. This comprehensive guide aims to provide a clear and concise explanation, suitable for students and anyone interested in enhancing their algebraic problem-solving abilities. By understanding the mechanics of solving quadratic equations, you can tackle a wide range of mathematical challenges with confidence and precision.

Understanding Quadratic Equations

Before we dive into solving the specific equation, let's establish a solid understanding of quadratic equations in general. A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is ax² + 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 represented by the quadratic equation intersects the x-axis. Understanding the nature of quadratic equations and their graphical representation is crucial for grasping the concepts involved in solving them. The coefficients a, b, and c play significant roles in determining the shape and position of the parabola, as well as the nature of the solutions. For instance, the sign of a determines whether the parabola opens upwards or downwards, and the discriminant (b² - 4ac) provides insights into the number and type of solutions (real or complex). Mastering these fundamental aspects of quadratic equations will pave the way for effectively solving a variety of problems in algebra and calculus.

Transforming the Equation

The first step in solving the equation 2x² - 5x + 1 = 3 is to transform it into the standard quadratic form, ax² + bx + c = 0. This involves subtracting 3 from both sides of the equation:2x² - 5x + 1 - 3 = 3 - 3. Simplifying this, we get 2x² - 5x - 2 = 0. Now the equation is in the standard form, where a = 2, b = -5, and c = -2. Transforming the equation into this standard form is crucial because it allows us to readily apply methods like the quadratic formula or factoring. The standard form provides a clear structure that facilitates the identification of the coefficients, which are essential for subsequent calculations. Furthermore, understanding this transformation process is a fundamental skill in algebra, as it allows for the manipulation and simplification of various types of equations. This step-by-step transformation not only prepares the equation for solving but also enhances one's algebraic intuition and problem-solving skills. Recognizing the importance of this initial step is key to successfully navigating more complex mathematical problems.

Applying the Quadratic Formula

To solve the quadratic equation 2x² - 5x - 2 = 0, we can use the quadratic formula. The quadratic formula is a powerful tool for finding the solutions of any quadratic equation in the form ax² + bx + c = 0. The formula is given by: x = (-b ± √(b² - 4ac)) / (2a). In our equation, a = 2, b = -5, and c = -2. Substituting these values into the quadratic formula, we get: x = (-(-5) ± √((-5)² - 4 * 2 * -2)) / (2 * 2). The quadratic formula is derived from the process of completing the square and provides a direct method for finding the roots of a quadratic equation. Its versatility lies in its ability to handle any quadratic equation, regardless of whether it can be easily factored or not. By plugging in the coefficients a, b, and c into the formula, we can obtain the solutions for x. The ± sign in the formula indicates that there are generally two solutions, corresponding to the two points where the parabola intersects the x-axis. Understanding the quadratic formula is essential for anyone studying algebra, as it provides a fundamental technique for solving a wide range of problems involving quadratic equations.

Calculating the Discriminant

Let's further simplify the equation by first calculating the discriminant. The discriminant is the part of the quadratic formula under the square root sign: b² - 4ac. In our case, b = -5, a = 2, and c = -2. So, the discriminant is (-5)² - 4 * 2 * (-2) = 25 + 16 = 41. The discriminant provides valuable information about the nature of the solutions. If the discriminant is positive, the equation has two distinct real solutions. If it is zero, the equation has one real solution (a repeated root). If it is negative, the equation has two complex solutions. In our case, the discriminant is 41, which is positive, indicating that the equation has two distinct real solutions. Understanding the discriminant is crucial for anticipating the type of solutions one might encounter when solving a quadratic equation. It helps in avoiding unnecessary calculations and provides a quick way to assess the solvability and nature of the roots. This concept is not only fundamental in algebra but also has applications in various fields such as physics and engineering.

Substituting the Discriminant Back into the Formula

Now that we have calculated the discriminant, we can substitute it back into the quadratic formula. We have: x = (5 ± √41) / 4. This expression gives us the two solutions for x. The ± sign indicates that there are two possible solutions: one obtained by adding the square root of 41 to 5 and then dividing by 4, and the other obtained by subtracting the square root of 41 from 5 and then dividing by 4. Substituting the discriminant back into the formula is a crucial step in obtaining the final solutions for x. It allows us to express the roots in terms of the coefficients of the quadratic equation and the discriminant. This process highlights the elegant relationship between the coefficients, the discriminant, and the solutions, showcasing the power and beauty of the quadratic formula. By understanding this substitution, one can efficiently and accurately determine the solutions of any quadratic equation, regardless of its complexity.

Final Solutions for x

Thus, the two solutions for x are: x = (5 + √41) / 4 and x = (5 - √41) / 4. These solutions correspond to option D: x = 5/4 ± √41 / 4. These are the values of x that satisfy the original equation, 2x² - 5x + 1 = 3. We have successfully solved the equation using the quadratic formula, a fundamental method in algebra. The solutions represent the points where the parabola represented by the quadratic equation intersects the x-axis. Understanding how to interpret these solutions graphically can provide a deeper understanding of the behavior of quadratic functions. Furthermore, the ability to find these solutions is crucial for solving a wide range of problems in mathematics, physics, engineering, and other fields. By mastering this technique, one can confidently tackle more complex mathematical challenges and apply this knowledge to real-world scenarios.

Conclusion

In conclusion, we have successfully solved the quadratic equation 2x² - 5x + 1 = 3 using the quadratic formula. The solutions are x = 5/4 ± √41 / 4, which corresponds to option D. This exercise demonstrates the power and versatility of the quadratic formula in solving quadratic equations. By understanding the steps involved, from transforming the equation into standard form to applying the formula and interpreting the results, one can confidently tackle a wide range of algebraic problems. The ability to solve quadratic equations is a fundamental skill in mathematics, with applications in various fields. Mastering this skill not only enhances one's mathematical proficiency but also provides a foundation for further studies in algebra, calculus, and beyond. The quadratic formula serves as a cornerstone in mathematical problem-solving, and its understanding is crucial for success in numerous scientific and engineering disciplines.