Solving 2x^2 + X - 1 = 2 Find The Solutions
In the realm of mathematics, solving quadratic equations is a fundamental skill. These equations, characterized by the highest power of the variable being 2, appear in various fields, from physics to engineering. This article delves into solving the specific quadratic equation 2x^2 + x - 1 = 2, providing a step-by-step guide and exploring different methods to arrive at the solutions x = -3/2 and x = 1. Understanding how to solve quadratic equations is crucial for anyone pursuing studies or careers involving quantitative analysis. This article aims to provide a clear and comprehensive explanation, suitable for both students and those seeking a refresher on the topic. We will cover the standard form of a quadratic equation, the quadratic formula, factoring techniques, and graphical solutions, ensuring a thorough understanding of the methods involved. Mastering these techniques will empower you to tackle a wide range of mathematical problems and real-world applications.
Before diving into the solution, it’s essential to understand what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is ax^2 + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'x' is the variable. The solutions to the quadratic equation are also known as the roots or zeros of the equation. These roots represent the values of 'x' that satisfy the equation. Quadratic equations are prevalent in various fields, including physics, engineering, economics, and computer science. They are used to model projectile motion, electrical circuits, financial markets, and many other phenomena. The coefficients 'a', 'b', and 'c' play crucial roles in determining the nature of the roots. The discriminant, given by the formula b^2 - 4ac, provides valuable information about the roots. If the discriminant is positive, there are two distinct real roots; if it is zero, there is one real root (or two equal real roots); and if it is negative, there are two complex roots. Understanding these fundamental concepts is crucial for effectively solving quadratic equations and applying them in various contexts.
The given equation, 2x^2 + x - 1 = 2, isn’t in the standard form required to directly apply methods like the quadratic formula or factoring. The first step is to transform the equation into the standard form ax^2 + bx + c = 0. To do this, subtract 2 from both sides of the equation: 2x^2 + x - 1 - 2 = 2 - 2. This simplifies to 2x^2 + x - 3 = 0. Now, the equation is in the standard form, where a = 2, b = 1, and c = -3. This standard form is essential because it allows us to easily identify the coefficients needed for various solution methods. The process of rearranging the equation into standard form is a crucial first step in solving any quadratic equation. It ensures that the equation is in a format that is compatible with the methods we will use to find the solutions. Failing to transform the equation into standard form can lead to errors in applying the quadratic formula or factoring techniques. Therefore, always make sure to rewrite the equation in the form ax^2 + bx + c = 0 before proceeding with further steps.
Factoring is a method of expressing the quadratic equation as a product of two binomials. This method is efficient when the quadratic equation can be factored easily. To factor the equation 2x^2 + x - 3 = 0, we need to find two numbers that multiply to (2 * -3 = -6) and add up to 1 (the coefficient of the x term). These two numbers are 3 and -2. Now, we rewrite the middle term using these numbers: 2x^2 + 3x - 2x - 3 = 0. Next, we factor by grouping: x(2x + 3) - 1(2x + 3) = 0. This simplifies to (x - 1)(2x + 3) = 0. For the product of two factors to be zero, at least one of the factors must be zero. Therefore, we set each factor equal to zero and solve for x:
- x - 1 = 0 => x = 1
- 2x + 3 = 0 => 2x = -3 => x = -3/2
Thus, the solutions are x = 1 and x = -3/2. Factoring is a powerful technique for solving quadratic equations, especially when the roots are rational numbers. However, not all quadratic equations can be factored easily. In such cases, other methods like the quadratic formula are more suitable. The key to successful factoring lies in identifying the correct pair of numbers that satisfy the conditions of multiplying to 'ac' and adding up to 'b'.
The quadratic formula is a universal method for solving quadratic equations of the form ax^2 + bx + c = 0. The formula is given by:
x = [-b ± √(b^2 - 4ac)] / (2a)
For the equation 2x^2 + x - 3 = 0, we have a = 2, b = 1, and c = -3. Substituting these values into the quadratic formula, we get:
x = [-1 ± √(1^2 - 4 * 2 * -3)] / (2 * 2)
x = [-1 ± √(1 + 24)] / 4
x = [-1 ± √25] / 4
x = [-1 ± 5] / 4
This gives us two possible solutions:
- x = (-1 + 5) / 4 = 4 / 4 = 1
- x = (-1 - 5) / 4 = -6 / 4 = -3/2
Thus, the solutions are x = 1 and x = -3/2, which matches the solutions obtained by factoring. The quadratic formula is a reliable method that works for all quadratic equations, regardless of whether they can be factored easily. It is particularly useful when dealing with equations that have irrational or complex roots. The formula is derived from the process of completing the square and provides a direct way to calculate the roots without the need for trial and error. Mastering the quadratic formula is an essential skill for anyone working with quadratic equations.
To ensure the accuracy of our solutions, it's crucial to verify them by substituting them back into the original equation, 2x^2 + x - 1 = 2.
Let’s first verify x = 1:
2(1)^2 + 1 - 1 = 2(1) + 1 - 1 = 2 + 1 - 1 = 2
The equation holds true for x = 1.
Now, let’s verify x = -3/2:
2(-3/2)^2 + (-3/2) - 1 = 2(9/4) - 3/2 - 1 = 9/2 - 3/2 - 1 = 6/2 - 1 = 3 - 1 = 2
Again, the equation holds true for x = -3/2. Since both values satisfy the original equation, we can confidently conclude that our solutions are correct. Verification is a critical step in the problem-solving process, particularly in mathematics. It helps to catch any errors made during the solution process and ensures that the final answers are accurate. By substituting the solutions back into the original equation, we can confirm that they indeed satisfy the equation's conditions. This step is especially important in exams and assignments, where accuracy is paramount.
In conclusion, we have successfully solved the quadratic equation 2x^2 + x - 1 = 2 using two different methods: factoring and the quadratic formula. Both methods yielded the same solutions: x = 1 and x = -3/2. We also verified these solutions by substituting them back into the original equation, confirming their correctness. Understanding and mastering various methods for solving quadratic equations is essential for mathematical proficiency. Factoring is a useful technique for equations that can be easily factored, while the quadratic formula provides a universal solution for all quadratic equations. The ability to solve quadratic equations is a fundamental skill that is applicable in various fields, including science, engineering, and economics. By understanding the underlying principles and practicing different solution methods, you can confidently tackle a wide range of mathematical problems. This article has provided a comprehensive guide to solving quadratic equations, equipping you with the knowledge and skills necessary to succeed in your mathematical endeavors. Remember to always verify your solutions to ensure accuracy and to choose the most appropriate method based on the specific equation you are solving.
