Solving 9x² + 36x + 36 = 0 A Step-by-Step Guide

Quadratic equations are a fundamental concept in algebra, and mastering their solutions is crucial for various mathematical and scientific applications. In this detailed guide, we will explore how to solve the specific quadratic equation 9x² + 36x + 36 = 0 using different methods, ensuring a thorough understanding of the underlying principles. Whether you're a student grappling with algebra or someone seeking a refresher, this article will provide a step-by-step approach to solving this equation, making the process clear and accessible.

Understanding Quadratic Equations

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, meaning the highest power of the variable (in this case, 'x') 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 zero. The constants a, b, and c are known as the coefficients of the quadratic equation. Understanding these coefficients is crucial for solving the equation effectively. For instance, in our equation, 9x² + 36x + 36 = 0, 'a' is 9, 'b' is 36, and 'c' is 36. Recognizing these coefficients is the first step towards applying the appropriate solution methods. Quadratic equations can have up to two distinct real roots, one real root (a repeated root), or no real roots, depending on the discriminant (b² - 4ac). The discriminant provides valuable information about the nature of the solutions, helping us anticipate the outcome before applying a solution method. Therefore, a solid grasp of the basic quadratic equation structure and its components sets the stage for solving complex problems with confidence.

Method 1: Factoring the Quadratic Equation

Factoring is often the quickest and most straightforward method for solving quadratic equations, but it requires the equation to be factorable. The goal is to rewrite the quadratic equation in the form (px + q)(rx + s) = 0, where p, q, r, and s are constants. To do this, we look for two numbers that multiply to 'ac' and add up to 'b'. In our equation, 9x² + 36x + 36 = 0, we can first simplify it by dividing all terms by the greatest common divisor, which is 9. This gives us x² + 4x + 4 = 0. Now, we need to find two numbers that multiply to 4 (1 * 4) and add up to 4. These numbers are 2 and 2. Thus, we can rewrite the equation as (x + 2)(x + 2) = 0, which can also be written as (x + 2)² = 0. Setting each factor equal to zero gives us x + 2 = 0, which leads to the solution x = -2. Since both factors are the same, we have a repeated root. Factoring is not always possible, especially when the roots are irrational or complex. In such cases, we turn to other methods like the quadratic formula. However, when factoring is applicable, it offers an elegant and efficient way to solve quadratic equations, making it an essential tool in any algebra toolkit.

Method 2: Using the Quadratic Formula

When factoring isn't feasible, the quadratic formula provides a reliable method for solving any quadratic equation. The quadratic formula is derived from the process of completing the square and is a universal solution for equations in the form ax² + bx + c = 0. The formula is given by: x = [-b ± √(b² - 4ac)] / (2a). To apply the quadratic formula to our equation, 9x² + 36x + 36 = 0, we first identify the coefficients: a = 9, b = 36, and c = 36. Plugging these values into the formula, we get: x = [-36 ± √(36² - 4 * 9 * 36)] / (2 * 9). Simplifying the expression under the square root, we have 36² - 4 * 9 * 36 = 1296 - 1296 = 0. This simplifies the formula to x = [-36 ± √0] / 18, which further simplifies to x = -36 / 18 = -2. The discriminant (b² - 4ac) being zero indicates that there is exactly one real root, which we have found to be x = -2. The quadratic formula not only provides the solution but also gives insight into the nature of the roots through the discriminant. A positive discriminant indicates two distinct real roots, a zero discriminant indicates one real root (a repeated root), and a negative discriminant indicates no real roots (complex roots). Therefore, mastering the quadratic formula is essential for solving any quadratic equation and understanding its solutions comprehensively.

Method 3: Completing the Square

Completing the square is another powerful method for solving quadratic equations, which involves transforming the equation into a perfect square trinomial. This method is particularly useful for understanding the structure of quadratic equations and can be used to derive the quadratic formula itself. To solve 9x² + 36x + 36 = 0 by completing the square, we first divide the entire equation by 9 to simplify it, resulting in x² + 4x + 4 = 0. The next step is to rewrite the equation in the form (x + k)² = m, where k and m are constants. To find 'k', we take half of the coefficient of the x term (which is 4) and square it. Half of 4 is 2, and 2 squared is 4. Notice that our simplified equation is already a perfect square trinomial: x² + 4x + 4 = (x + 2)². Thus, we can rewrite the equation as (x + 2)² = 0. Taking the square root of both sides, we get x + 2 = 0, which gives us the solution x = -2. Completing the square provides a clear, step-by-step approach to solving quadratic equations, and it's especially valuable for understanding the relationship between the roots and the coefficients of the equation. This method can also be used to solve equations where the leading coefficient is not 1, although it may involve additional steps to ensure the coefficient of x² is 1 before proceeding. Mastering completing the square enhances problem-solving skills and provides a deeper understanding of quadratic equations.

Verifying the Solution

After solving a quadratic equation, it's crucial to verify the solution to ensure accuracy. This step is essential for preventing errors and building confidence in your problem-solving skills. To verify our solution, x = -2, for the equation 9x² + 36x + 36 = 0, we substitute -2 for x in the original equation. This gives us 9(-2)² + 36(-2) + 36. Evaluating this expression, we have 9(4) - 72 + 36, which simplifies to 36 - 72 + 36. This further simplifies to 72 - 72, which equals 0. Since the equation holds true when x = -2, we can confidently say that our solution is correct. Verification is not just a formality; it's a critical step in the problem-solving process that reinforces understanding and minimizes mistakes. Whether you've used factoring, the quadratic formula, or completing the square, always take the time to plug your solution back into the original equation. This practice will strengthen your algebraic skills and ensure you arrive at the correct answer.

Conclusion

In conclusion, solving the quadratic equation 9x² + 36x + 36 = 0 can be achieved through various methods, each providing a unique approach to the problem. We explored factoring, which allowed us to simplify the equation to (x + 2)² = 0, leading to the solution x = -2. We also utilized the quadratic formula, a universal method that confirmed our solution and provided insight into the nature of the roots. Additionally, completing the square offered a step-by-step transformation of the equation, reinforcing our understanding of its structure. Verifying the solution by substituting x = -2 back into the original equation ensured the accuracy of our result. Mastering these methods equips you with a comprehensive toolkit for solving quadratic equations and tackling more complex algebraic problems. The ability to choose the most efficient method for a given equation, whether it's factoring for simpler cases or the quadratic formula for more complex ones, is a key skill in mathematics. By practicing and understanding these techniques, you can confidently approach and solve a wide range of quadratic equations.