Solutions To The Equation 0 = X² - X - 6

Understanding quadratic equations is crucial in algebra, and finding their solutions, also known as roots, is a fundamental skill. In this article, we will delve into the process of solving the quadratic equation 0 = x² - x - 6. We will explore different methods to find the values of x that satisfy this equation and discuss the correct options from the given choices. This comprehensive guide aims to provide a clear and detailed explanation, ensuring you grasp the concepts thoroughly. Solving quadratic equations is not only a core concept in mathematics but also has practical applications in various fields such as physics, engineering, and economics. Therefore, mastering this skill is essential for anyone pursuing studies or careers in these areas. By understanding the techniques involved in solving quadratic equations, you can tackle more complex mathematical problems and real-world scenarios with confidence. In this particular instance, we are presented with a specific quadratic equation, 0 = x² - x - 6, and a set of potential solutions. Our task is to identify the values of x that make the equation true. We will accomplish this by employing methods such as factoring, the quadratic formula, and other algebraic techniques. Each method offers a unique approach to solving the equation, and understanding these approaches will enhance your problem-solving abilities. Let's begin by examining the equation more closely and exploring the most effective strategies for finding its roots. The goal is to not only find the solutions but also to understand the underlying principles that govern quadratic equations. This understanding will empower you to solve a wide range of similar problems in the future. So, let's embark on this mathematical journey together and unlock the solutions to the equation 0 = x² - x - 6.

Methods to Solve Quadratic Equations

To solve the quadratic equation 0 = x² - x - 6, we can employ several methods, each with its own strengths and applications. Two primary approaches are factoring and using the quadratic formula. Factoring involves breaking down the quadratic expression into a product of two binomials, while the quadratic formula provides a direct solution for any quadratic equation in the standard form ax² + bx + c = 0. In this section, we will explore both methods in detail, demonstrating how they can be applied to find the roots of the given equation. Understanding these methods is crucial for solving a wide range of quadratic equations and will provide you with a versatile toolkit for algebraic problem-solving. Factoring is often the preferred method when the quadratic expression can be easily factored, as it tends to be quicker and more intuitive. However, not all quadratic equations can be factored easily, which is where the quadratic formula becomes invaluable. The quadratic formula is a universal method that can be applied to any quadratic equation, regardless of its factorability. It is particularly useful when dealing with equations that have irrational or complex roots. By mastering both factoring and the quadratic formula, you will be well-equipped to tackle any quadratic equation that comes your way. Additionally, understanding these methods will deepen your comprehension of quadratic equations and their properties. Let's delve into each method, starting with factoring, and see how they can be used to solve our equation 0 = x² - x - 6.

Factoring the Quadratic Equation

Factoring is a powerful technique to solve quadratic equations. Our equation is 0 = x² - x - 6. To factor this, we look for two numbers that multiply to -6 (the constant term) and add up to -1 (the coefficient of the x term). These numbers are -3 and 2, since (-3) * 2 = -6 and (-3) + 2 = -1. Therefore, we can rewrite the equation as:

0 = (x - 3)(x + 2)

To find the solutions, we set each factor equal to zero:

x - 3 = 0 or x + 2 = 0

Solving these linear equations gives us:

x = 3 or x = -2

Thus, the solutions to the quadratic equation 0 = x² - x - 6 are x = 3 and x = -2. Factoring is an efficient method for solving quadratic equations when the roots are integers or simple fractions. It relies on the principle that if the product of two factors is zero, then at least one of the factors must be zero. This allows us to break down the quadratic equation into two simpler linear equations, which are easy to solve. The key to successful factoring is to identify the correct pair of numbers that satisfy the multiplication and addition conditions. This often involves some trial and error, but with practice, you can become proficient at recognizing common factoring patterns. Factoring not only provides a solution to the equation but also gives insight into the structure of the quadratic expression. It reveals the roots of the equation and can be used to graph the corresponding parabola. In this case, factoring has provided us with two clear solutions, x = 3 and x = -2, which we can now compare with the given options.

Using the Quadratic Formula

The quadratic formula is a universal method for solving quadratic equations of the form ax² + bx + c = 0. The formula is:

x = (-b ± √(b² - 4ac)) / (2a)

For our equation, 0 = x² - x - 6, we have a = 1, b = -1, and c = -6. Plugging these values into the formula, we get:

x = (-(-1) ± √((-1)² - 4 * 1 * -6)) / (2 * 1)

x = (1 ± √(1 + 24)) / 2

x = (1 ± √25) / 2

x = (1 ± 5) / 2

This gives us two solutions:

x = (1 + 5) / 2 = 6 / 2 = 3

x = (1 - 5) / 2 = -4 / 2 = -2

So, the solutions are x = 3 and x = -2, which matches our results from factoring. The quadratic formula is a powerful tool because it can be used to solve any quadratic equation, regardless of whether it can be factored easily. It is particularly useful when the roots are irrational or complex numbers. The formula is derived from the process of completing the square and provides a direct method for finding the roots. While it may seem more complex than factoring, it is a reliable method that always yields the correct solutions. The key to using the quadratic formula effectively is to correctly identify the coefficients a, b, and c and substitute them into the formula. Care must be taken to handle the signs and perform the arithmetic operations accurately. The discriminant, b² - 4ac, within the formula, provides valuable information about the nature of the roots. If the discriminant is positive, the equation has two distinct real roots. If it is zero, the equation has one real root (a repeated root). If it is negative, the equation has two complex roots. In our case, the discriminant is 25, which is positive, indicating two distinct real roots. The quadratic formula confirms the solutions we obtained through factoring, reinforcing the accuracy of our results.

Identifying the Correct Options

Now that we have solved the quadratic equation 0 = x² - x - 6 using both factoring and the quadratic formula, we have determined that the solutions are x = 3 and x = -2. Let's compare these solutions with the given options:

A. x = -3 B. x = -2 C. x = 0 D. x = 2 E. x = 3

By comparing our solutions with the options, we can see that options B (x = -2) and E (x = 3) are the correct answers. This step is crucial to ensure that we are selecting the appropriate solutions from the given choices. It involves careful comparison and verification to avoid any errors. It's also a good practice to plug the solutions back into the original equation to confirm their validity. This provides an additional layer of assurance that the solutions are correct. In this case, substituting x = -2 and x = 3 into the equation 0 = x² - x - 6, we can verify that both values satisfy the equation. This process of verification is an important aspect of problem-solving in mathematics and helps to build confidence in the correctness of the solutions. Now that we have identified the correct options, we can confidently conclude that the solutions to the equation are x = -2 and x = 3. The process of solving a quadratic equation involves several steps, from choosing the appropriate method to verifying the solutions. By following these steps carefully, we can arrive at the correct answers and gain a deeper understanding of quadratic equations.

Conclusion

In conclusion, the solutions to the quadratic equation 0 = x² - x - 6 are x = -2 and x = 3. We arrived at these solutions by employing two different methods: factoring and the quadratic formula. Factoring involved breaking down the quadratic expression into two binomial factors, while the quadratic formula provided a direct calculation of the roots. Both methods yielded the same results, reinforcing the correctness of our solutions. This exercise highlights the importance of understanding different problem-solving techniques and choosing the most appropriate method for a given situation. Factoring is often a quicker method when the equation can be easily factored, while the quadratic formula is a universal method that can be applied to any quadratic equation. By mastering both methods, you can tackle a wide range of quadratic equations with confidence. Furthermore, understanding the underlying principles of quadratic equations is crucial for success in mathematics and related fields. Quadratic equations appear in various contexts, from physics and engineering to economics and finance. The ability to solve these equations is a valuable skill that can be applied to real-world problems. This article has provided a detailed explanation of the process of solving a specific quadratic equation, but the principles and techniques discussed can be applied to many other problems. By practicing these methods and deepening your understanding of quadratic equations, you can enhance your problem-solving abilities and achieve success in your mathematical endeavors. Remember to always verify your solutions and strive for a thorough understanding of the concepts involved. With practice and dedication, you can master the art of solving quadratic equations and other mathematical challenges.