Solving X² + 2x = 24 A Step By Step Guide

In the realm of algebra, solving equations is a fundamental skill. Quadratic equations, in particular, hold a significant place due to their frequent appearance in various mathematical and real-world applications. This article delves into the process of solving a specific quadratic equation, x² + 2x = 24, exploring the steps involved in finding the values of x that satisfy the equation. Our main keyword, solving quadratic equations, will be central to our discussion. We aim to provide a comprehensive guide, ensuring that readers can understand and apply these techniques to similar problems.

The importance of understanding quadratic equations extends beyond the classroom. They are used in physics to model projectile motion, in engineering to design structures, and in economics to analyze market trends. The ability to solve these equations is therefore a valuable asset in a variety of fields. This article will not only focus on the mechanical steps of solving the equation but also emphasize the underlying principles and concepts. By doing so, we hope to empower readers to tackle a wide range of quadratic equation problems with confidence and competence.

Before we dive into the specifics of our example, let's briefly touch on the general form of a quadratic equation: ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. Our equation, x² + 2x = 24, can be transformed into this standard form, which is a crucial first step in the solving process. The methods we will explore, such as factoring and using the quadratic formula, are applicable to any quadratic equation in standard form. This article aims to demystify these methods and make them accessible to all readers, regardless of their prior experience with algebra. Our discussion will be enriched with examples and explanations to enhance understanding and retention. We believe that a solid grasp of quadratic equations is essential for anyone pursuing further studies in mathematics or related fields.

The first crucial step in solving quadratic equations like x² + 2x = 24 is to transform it into the standard form, which is ax² + bx + c = 0. This form is essential because it allows us to apply various methods, such as factoring or the quadratic formula, to find the solutions for x. In our equation, we have x² + 2x = 24. To get it into the standard form, we need to move the constant term, 24, to the left side of the equation. This is achieved by subtracting 24 from both sides of the equation. This process maintains the equality and rearranges the equation into a more manageable format. The resulting equation is x² + 2x - 24 = 0. This is now a standard quadratic equation where we can clearly identify the coefficients a, b, and c. In this case, a = 1, b = 2, and c = -24. Identifying these coefficients is vital for applying the quadratic formula or for factoring the equation.

The importance of the standard form cannot be overstated. It provides a consistent framework for solving quadratic equations, regardless of their initial appearance. Without this standardization, the solving process would be much more complex and less systematic. The standard form also helps in visualizing the relationship between the coefficients and the solutions of the equation. For instance, the sign of the constant term c can provide insights into the nature of the roots (solutions) of the equation. In our example, c is negative, which suggests that the equation will have two real roots with opposite signs. This is a useful observation that can guide our solving strategy. Furthermore, the standard form is a prerequisite for using the quadratic formula, a powerful tool that can solve any quadratic equation, even those that are difficult to factor. Thus, mastering the transformation to standard form is a fundamental skill in algebra and a cornerstone of solving quadratic equations effectively.

Once the equation is in standard form, we can proceed with various methods to find the values of x. The choice of method often depends on the specific characteristics of the equation. Some equations are easily factored, while others require the use of the quadratic formula. Understanding the strengths and limitations of each method is crucial for efficient problem-solving. In the following sections, we will explore factoring as a method to solve our example equation, x² + 2x - 24 = 0. Factoring involves breaking down the quadratic expression into two binomial factors, which can then be used to find the roots of the equation. This method relies on the relationship between the coefficients of the equation and the factors. It is a powerful technique when applicable and provides a deeper understanding of the structure of quadratic equations. The ability to transform a quadratic equation into standard form is, therefore, a foundational skill that unlocks a range of problem-solving approaches in algebra.

Factoring is a powerful technique for solving quadratic equations, especially when the equation can be expressed as a product of two binomials. In our case, we have the equation x² + 2x - 24 = 0. The goal of factoring is to find two binomials that, when multiplied together, give us the original quadratic expression. To do this, we look for two numbers that multiply to give the constant term (-24) and add up to the coefficient of the x term (2). This is a crucial step, and finding the correct pair of numbers is key to successful factoring.

Let's consider the factors of -24. We have pairs like (1, -24), (-1, 24), (2, -12), (-2, 12), (3, -8), (-3, 8), (4, -6), and (-4, 6). Among these pairs, we are looking for the one that adds up to 2. The pair (-4, 6) satisfies this condition because -4 * 6 = -24 and -4 + 6 = 2. Therefore, we can rewrite the quadratic expression as (x - 4)(x + 6) = 0. This factored form of the equation is equivalent to the original equation, but it provides a much clearer path to finding the solutions for x.

The beauty of factoring lies in its simplicity and elegance. It transforms a complex problem into a straightforward one. Once we have the equation in factored form, we can use the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. In other words, if ab = 0, then either a = 0 or b = 0 (or both). Applying this property to our equation (x - 4)(x + 6) = 0, we can set each factor equal to zero and solve for x. This gives us two separate equations: x - 4 = 0 and x + 6 = 0. Solving the first equation, we add 4 to both sides, which gives us x = 4. Solving the second equation, we subtract 6 from both sides, which gives us x = -6. These two values, x = 4 and x = -6, are the solutions to our quadratic equation. They are the values of x that make the equation x² + 2x = 24 true. Factoring, when applicable, provides a direct and intuitive way to find these solutions.

After solving quadratic equations, it's crucial to verify the solutions to ensure accuracy. This step helps catch any errors made during the solving process and confirms that the values obtained truly satisfy the original equation. In our case, we found two potential solutions for the equation x² + 2x = 24: x = 4 and x = -6. To verify these solutions, we substitute each value back into the original equation and check if the equation holds true. This process is a fundamental part of problem-solving in mathematics, reinforcing the understanding of what it means for a value to be a solution to an equation.

Let's first verify x = 4. Substituting this value into the equation x² + 2x = 24, we get (4)² + 2(4) = 16 + 8 = 24. Since the left side of the equation equals the right side (24 = 24), the solution x = 4 is verified. This confirms that when x is 4, the equation holds true. Now, let's verify x = -6. Substituting this value into the equation x² + 2x = 24, we get (-6)² + 2(-6) = 36 - 12 = 24. Again, the left side of the equation equals the right side (24 = 24), so the solution x = -6 is also verified. This demonstrates that when x is -6, the equation also holds true.

The importance of verification extends beyond simply checking for errors. It deepens the understanding of the relationship between the solutions and the original equation. By substituting the solutions back into the equation, we gain a concrete sense of how these values make the equation balanced. This process is particularly valuable when dealing with more complex equations or systems of equations, where the potential for errors is higher. Verification also provides confidence in the correctness of the solutions, which is crucial in applications where the solutions are used as inputs for further calculations or decisions. In summary, verifying solutions is an essential step in the problem-solving process, ensuring accuracy, reinforcing understanding, and building confidence in mathematical skills.

In this article, we've thoroughly explored the process of solving quadratic equations using the example x² + 2x = 24. We began by transforming the equation into its standard form, x² + 2x - 24 = 0, which is a crucial first step for applying various solving methods. We then employed the factoring technique, identifying the factors (x - 4)(x + 6) = 0. By applying the zero-product property, we found two potential solutions: x = 4 and x = -6. Finally, we verified these solutions by substituting them back into the original equation, confirming their validity. This step-by-step approach highlights the systematic nature of solving quadratic equations and the importance of each stage in the process.

The skills and concepts covered in this article are fundamental to algebra and have broad applications in various fields. Understanding how to transform equations into standard form, factor quadratic expressions, and verify solutions are essential tools for anyone studying mathematics or related disciplines. The ability to solve quadratic equations is not just an academic exercise; it's a practical skill that can be applied to real-world problems in physics, engineering, economics, and other areas. This article aimed to provide a clear and comprehensive guide, empowering readers to confidently tackle similar problems.

In conclusion, solving quadratic equations is a fundamental skill in mathematics, and this article has provided a detailed walkthrough of the process using a specific example. By understanding the steps involved and practicing these techniques, readers can develop a strong foundation in algebra and enhance their problem-solving abilities. The solutions to the equation x² + 2x = 24 are x = 4 and x = -6, which were found through factoring and subsequently verified. This process underscores the importance of a systematic approach and the value of verifying solutions to ensure accuracy. The knowledge and skills gained from this article will undoubtedly be beneficial for further studies in mathematics and its applications.