Solving X² - 36 = 0 Step-by-Step Guide

Jul 9, 2025 by ADMIN 39 views

Solving the Quadratic Equation: x² - 36 = 0

In the realm of mathematics, solving equations is a fundamental skill. This article delves into the process of solving a specific quadratic equation, x² - 36 = 0. We'll break down the steps, explore different methods, and arrive at the correct solutions. Understanding how to solve quadratic equations is crucial for various mathematical and scientific applications, ranging from physics problems to engineering calculations. Therefore, mastering this concept is essential for anyone pursuing studies or careers in these fields. We will explore this equation not just to find the solutions, but also to understand the underlying principles and methods that can be applied to a broader range of quadratic equations. This comprehensive approach ensures a deeper understanding and the ability to tackle more complex problems in the future. Furthermore, we'll discuss the significance of the solutions in the context of quadratic equations and their graphical representation. This will provide a more holistic view of the equation and its properties, enhancing your mathematical intuition and problem-solving skills. Ultimately, this guide aims to equip you with the knowledge and confidence to solve quadratic equations effectively and efficiently.

Understanding Quadratic Equations

Before diving into the solution, let's define what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable is 2. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants and a is not equal to 0. In our case, the equation x² - 36 = 0 fits this form, with a = 1, b = 0, and c = -36. The absence of the x term (i.e., bx) simplifies the equation, making it easier to solve using specific methods. However, it's important to recognize the general form to understand the broader context of quadratic equations and their solutions. Quadratic equations arise in numerous real-world applications, such as modeling projectile motion, calculating areas and volumes, and optimizing various processes. Therefore, a solid understanding of quadratic equations is crucial for anyone working in STEM fields. Furthermore, the solutions to quadratic equations, also known as roots, can provide valuable insights into the behavior of the corresponding quadratic function, which is a parabola when graphed. These roots represent the points where the parabola intersects the x-axis, and their properties can be used to analyze the function's characteristics.

Method 1: Factoring

The equation x² - 36 = 0 can be solved using the factoring method, which relies on recognizing the difference of squares pattern. The difference of squares pattern states that a² - b² = (a + b)(a - b). In our equation, x² corresponds to a² and 36 corresponds to b², since 36 is 6 squared. Applying the difference of squares pattern, we can factor the equation as follows: x² - 36 = (x + 6)(x - 6) = 0. Now, for the product of two factors to be zero, at least one of the factors must be zero. Therefore, we have two possible cases: x + 6 = 0 or x - 6 = 0. Solving for x in each case, we get x = -6 and x = 6. These are the two solutions to the quadratic equation. Factoring is a powerful technique for solving quadratic equations, especially when the equation can be easily factored using patterns like the difference of squares. It's a direct and efficient method that provides a clear understanding of the solutions. Moreover, factoring reinforces the connection between algebraic expressions and their factored forms, which is a fundamental concept in algebra. By mastering factoring, you can simplify complex equations and gain insights into their underlying structure. This method is particularly useful when the coefficients of the quadratic equation are integers, making the factoring process more straightforward.

Method 2: Square Root Property

Another method to solve x² - 36 = 0 is using the square root property. This method is particularly effective when the equation is in the form x² = c, where c is a constant. To apply this method, we first isolate the x² term: x² = 36. Now, we take the square root of both sides of the equation, remembering that the square root of a number can be positive or negative. This gives us x = ±√36. Since the square root of 36 is 6, we have x = ±6, which means x = 6 or x = -6. This method is concise and direct, making it a preferred choice for equations in this specific form. The square root property highlights the inverse relationship between squaring a number and taking its square root, which is a fundamental concept in algebra. By understanding this relationship, you can efficiently solve equations where the variable is squared. Furthermore, the square root property emphasizes the importance of considering both positive and negative solutions when taking the square root of a number. This is crucial for obtaining all possible solutions to the equation and ensuring a complete understanding of the problem. The square root property is a valuable tool in your mathematical arsenal, and its application can significantly simplify the process of solving quadratic equations.

The Solutions

Both the factoring method and the square root property lead us to the same solutions: x = 6 and x = -6. Therefore, the correct answer is C. x = -6 ; x = 6. These solutions represent the values of x that make the equation x² - 36 = 0 true. In the context of a quadratic function f(x) = x² - 36, these solutions are the x-intercepts, the points where the graph of the function crosses the x-axis. Understanding the solutions not only provides the numerical answers but also gives insights into the behavior of the quadratic function. The solutions are symmetrically located around the y-axis, reflecting the symmetry of the parabola represented by the function. This symmetry is a characteristic feature of quadratic functions and can be used to analyze their properties. Moreover, the solutions can be used to determine the vertex of the parabola, which is the point where the function reaches its minimum or maximum value. The vertex is located at the midpoint of the solutions, in this case, at x = 0. By analyzing the solutions and their relationship to the function's graph, you can gain a deeper understanding of quadratic equations and their applications.

Why Other Options are Incorrect

Let's briefly examine why the other answer options are incorrect:

A. x = 1 ; x = -36: Substituting these values into the equation x² - 36 = 0 yields 1² - 36 = -35 ≠ 0 and (-36)² - 36 = 1260 ≠ 0. Thus, these are not solutions.
B. x = -1 ; x = 36: Substituting these values into the equation x² - 36 = 0 yields (-1)² - 36 = -35 ≠ 0 and (36)² - 36 = 1260 ≠ 0. Thus, these are not solutions.
D. x = -18 ; x = 18: Substituting these values into the equation x² - 36 = 0 yields (-18)² - 36 = 288 ≠ 0 and (18)² - 36 = 288 ≠ 0. Thus, these are not solutions.

These incorrect options highlight the importance of verifying solutions by substituting them back into the original equation. This step ensures that the calculated values indeed satisfy the equation and are not extraneous solutions. By carefully checking the solutions, you can avoid errors and gain confidence in your answers. Moreover, analyzing the incorrect options can help you understand common mistakes and develop a more robust problem-solving approach. It's crucial to not only find the correct solutions but also to understand why other options are incorrect. This comprehensive understanding will enhance your mathematical skills and prevent you from making similar errors in the future. Always remember to verify your solutions and analyze the incorrect options to solidify your knowledge of the subject.

Conclusion

In conclusion, we have successfully solved the quadratic equation x² - 36 = 0 using two different methods: factoring and the square root property. Both methods led us to the solutions x = 6 and x = -6. Understanding these methods and their applications is crucial for solving a wide range of quadratic equations. Mastering these techniques will not only improve your problem-solving skills but also enhance your understanding of the fundamental principles of algebra. Furthermore, the ability to solve quadratic equations is essential for various mathematical and scientific disciplines, making it a valuable skill for academic and professional pursuits. By practicing these methods and applying them to different types of equations, you can develop a strong foundation in algebra and build confidence in your mathematical abilities. Remember to always verify your solutions and analyze the problem from different perspectives to gain a deeper understanding of the concepts involved. With consistent effort and practice, you can master the art of solving quadratic equations and unlock the power of mathematics.