Solving Quadratic Equations The Roots Of 2x² - 5x - 3 = 0

This article delves into the process of finding the roots of the quadratic equation 2x² - 5x - 3 = 0. Understanding how to solve quadratic equations is a fundamental skill in algebra, with applications spanning various fields of mathematics and beyond. We will explore different methods to solve this equation, providing a comprehensive guide for students and enthusiasts alike. The correct answer is B) -1/2 and 3. This article will provide a step-by-step explanation of how to arrive at this solution.

Understanding Quadratic Equations

Quadratic equations are polynomial equations 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 roots of a quadratic equation are the values of x that satisfy the equation, also known as the solutions or zeros of the equation. Finding these roots is a common problem in algebra, and there are several methods to tackle it. In this case, our quadratic equation is 2x² - 5x - 3 = 0, where a = 2, b = -5, and c = -3. To find the roots, we need to determine the values of x that make this equation true. There are primarily three methods to solve quadratic equations: factoring, using the quadratic formula, and completing the square. We will focus on the factoring method and the quadratic formula in this article, as they are the most commonly used techniques. Factoring involves expressing the quadratic expression as a product of two binomials. If we can factor the equation, we can easily find the roots by setting each factor equal to zero and solving for x. The quadratic formula, on the other hand, provides a direct way to find the roots using the coefficients a, b, and c. It is particularly useful when the equation is difficult to factor. Both methods are powerful tools for solving quadratic equations, and mastering them is crucial for success in algebra and related fields. Understanding the nature of quadratic equations and their roots is also essential for various applications, such as modeling physical phenomena, solving optimization problems, and analyzing data. By exploring these concepts in detail, we aim to provide a comprehensive understanding of how to solve quadratic equations effectively.

Method 1: Factoring the Quadratic Equation

Factoring is often the quickest way to solve a quadratic equation if the equation can be factored easily. To factor the quadratic equation 2x² - 5x - 3 = 0, we need to find two binomials that, when multiplied, give us the original quadratic expression. This involves breaking down the quadratic expression into two linear factors. The process requires careful consideration of the coefficients and the constant term. First, we look for two numbers that multiply to give the product of the leading coefficient (2) and the constant term (-3), which is -6. At the same time, these two numbers must add up to the middle coefficient, which is -5. The two numbers that satisfy these conditions are -6 and 1 because (-6) * (1) = -6 and (-6) + (1) = -5. Next, we rewrite the middle term (-5x) using these two numbers: 2x² - 6x + 1x - 3 = 0. This step is crucial as it allows us to group terms and factor by grouping. Now, we group the first two terms and the last two terms: (2x² - 6x) + (1x - 3) = 0. From the first group, we can factor out 2x, and from the second group, we can factor out 1: 2x(x - 3) + 1(x - 3) = 0. Notice that both terms now have a common factor of (x - 3). We can factor this out: (2x + 1)(x - 3) = 0. Now we have the quadratic equation factored into two binomials. To find the roots, we set each factor equal to zero: 2x + 1 = 0 and x - 3 = 0. Solving the first equation, 2x + 1 = 0, we subtract 1 from both sides: 2x = -1. Then, we divide by 2: x = -1/2. Solving the second equation, x - 3 = 0, we add 3 to both sides: x = 3. Therefore, the roots of the quadratic equation 2x² - 5x - 3 = 0 are -1/2 and 3. This method demonstrates the power of factoring in simplifying quadratic equations and finding their solutions. Factoring is a fundamental skill in algebra and is often the most efficient method when the quadratic expression can be easily factored. Understanding the process of factoring and practicing with different examples can greatly improve problem-solving skills in mathematics.

Method 2: Using the Quadratic Formula

When factoring is not straightforward or possible, the quadratic formula provides a reliable method for finding the roots of a quadratic equation. The quadratic formula is derived from the method of completing the square and is a general solution for any quadratic equation in the form ax² + bx + c = 0. The formula is given by: x = (-b ± √(b² - 4ac)) / (2a). This formula allows us to directly calculate the roots of the equation using the coefficients a, b, and c. For our equation, 2x² - 5x - 3 = 0, we have a = 2, b = -5, and c = -3. Plugging these values into the quadratic formula, we get: x = (-(-5) ± √((-5)² - 4 * 2 * (-3))) / (2 * 2). Simplify the expression inside the square root: (-5)² = 25 and 4 * 2 * (-3) = -24. So, the expression becomes: x = (5 ± √(25 - (-24))) / 4. Further simplifying, we have: x = (5 ± √(25 + 24)) / 4. This simplifies to: x = (5 ± √49) / 4. The square root of 49 is 7, so we get: x = (5 ± 7) / 4. Now, we have two possible solutions, one with the plus sign and one with the minus sign. For the plus sign: x = (5 + 7) / 4 = 12 / 4 = 3. For the minus sign: x = (5 - 7) / 4 = -2 / 4 = -1/2. Therefore, the roots of the equation 2x² - 5x - 3 = 0, using the quadratic formula, are 3 and -1/2. This result matches the solution we obtained through factoring, confirming the correctness of both methods. The quadratic formula is a powerful tool because it can be applied to any quadratic equation, regardless of whether it can be factored easily. It is particularly useful when the roots are not rational numbers, as factoring may not be feasible in such cases. Understanding and applying the quadratic formula is an essential skill in algebra and is widely used in various mathematical and scientific applications. Mastering this formula allows for the efficient and accurate solution of quadratic equations.

Comparing the Methods

Both factoring and the quadratic formula are effective methods for finding the roots of a quadratic equation, but they have different strengths and weaknesses. Factoring is generally quicker and simpler when the quadratic expression can be easily factored. It involves breaking down the expression into two binomials, which can be a straightforward process if the coefficients and constant term allow for integer roots. However, factoring can be challenging or even impossible when the roots are irrational or complex numbers. In such cases, the quadratic formula is the more reliable method. The quadratic formula is a general solution that can be applied to any quadratic equation, regardless of the nature of its roots. It provides a direct way to calculate the roots using the coefficients of the equation. While the formula itself is straightforward, it can involve more calculations than factoring, especially when simplifying the expression under the square root. In the case of the equation 2x² - 5x - 3 = 0, both methods can be used effectively. Factoring is relatively straightforward, as we found two binomials (2x + 1) and (x - 3) that multiply to give the original expression. The quadratic formula also yields the same roots, providing a confirmation of our solution. When choosing between the two methods, it's important to consider the specific equation and the ease with which it can be factored. If factoring is quick and easy, it is often the preferred method. However, if factoring seems difficult or if the roots are expected to be irrational, the quadratic formula is the more reliable choice. Ultimately, mastering both methods is beneficial, as it provides flexibility and the ability to solve a wide range of quadratic equations. Understanding the strengths and weaknesses of each method allows for a more strategic approach to problem-solving in algebra.

Conclusion

In conclusion, we have successfully found the roots of the quadratic equation 2x² - 5x - 3 = 0 using two different methods: factoring and the quadratic formula. The roots, as we determined, are -1/2 and 3. Both methods are valuable tools in algebra, and understanding when and how to use them is crucial for solving quadratic equations efficiently. Factoring is often the quicker method when the equation can be easily factored, while the quadratic formula provides a reliable solution for any quadratic equation, regardless of its factorability. Mastering these techniques not only enhances problem-solving skills in mathematics but also lays a strong foundation for more advanced topics. Quadratic equations are fundamental in various fields, including physics, engineering, and economics, making their understanding essential for students and professionals alike. By exploring different methods and comparing their strengths and weaknesses, we gain a deeper appreciation for the versatility and power of algebraic tools. In this particular case, both methods led us to the same correct answer, reinforcing the accuracy and consistency of these approaches. As we continue to explore mathematics, the ability to solve quadratic equations effectively will undoubtedly prove to be a valuable asset. The process of finding roots is not just about arriving at the correct answer; it's also about understanding the underlying principles and developing a strategic approach to problem-solving. By mastering these fundamental concepts, we can confidently tackle more complex mathematical challenges and apply our knowledge to real-world situations.