Solving Quadratic Equations Completing The Square Method

Introduction

In this comprehensive guide, we will delve into the method of completing the square to solve quadratic equations. This technique is a powerful tool in algebra, allowing us to find the roots of any quadratic equation, regardless of whether it can be easily factored. Our example equation, 2x² + 2x + 49 = 4x + 170, presented by Edna Ann R. Cavalida from Grade 9 - Cherry Blossom, serves as a practical case to illustrate the step-by-step process. Understanding the method of completing the square is crucial for students and anyone seeking a deeper understanding of quadratic equations. This article aims to not only solve the given equation but also to provide a thorough explanation of the underlying principles and steps involved in this method. By the end of this guide, you will be equipped with the knowledge and skills to tackle a wide range of quadratic equations using the completing the square technique. The method not only helps in finding solutions but also provides insights into the structure and properties of quadratic expressions. This method is particularly useful when the quadratic equation cannot be easily factored, making it a versatile tool in mathematical problem-solving. The steps involved in completing the square are systematic and logical, which enhances mathematical reasoning and problem-solving skills.

Identifying the Problem

The initial step in solving any quadratic equation is to identify the problem clearly. In our case, the equation is given as 2x² + 2x + 49 = 4x + 170. This equation is a quadratic equation because it contains a term with x squared (x²). The goal is to find the values of x that satisfy this equation. To solve it using the completing the square method, we need to rearrange the equation into a standard form and then apply the steps systematically. The first step involves simplifying the equation by moving all terms to one side, resulting in an equation equal to zero. This process is crucial as it sets the stage for the subsequent steps in completing the square. Identifying the problem also involves recognizing the coefficients of the quadratic, linear, and constant terms, which are essential for applying the completing the square method correctly. In our equation, we need to identify the coefficients of x², x, and the constant terms on both sides of the equation before rearranging it into the standard form. This initial identification is a critical step in ensuring that the subsequent algebraic manipulations are accurate and lead to the correct solutions. By focusing on this methodical approach, we can avoid common errors and develop a stronger understanding of the problem-solving process in algebra.

Rewriting the Equation in Standard Form

Before we can apply the completing the square method, we need to rewrite the given equation, 2x² + 2x + 49 = 4x + 170, into the standard quadratic form, which is Ax² + Bx + C = 0. This form is essential because it allows us to easily identify the coefficients needed for the subsequent steps in the completing the square method. To achieve this standard form, we need to move all terms to one side of the equation, leaving zero on the other side. Let's start by subtracting 4x from both sides of the equation: 2x² + 2x - 4x + 49 = 170. Next, subtract 170 from both sides: 2x² + 2x - 4x + 49 - 170 = 0. Now, we simplify the equation by combining like terms. The x terms can be combined as 2x - 4x = -2x, and the constant terms can be combined as 49 - 170 = -121. This gives us the equation 2x² - 2x - 121 = 0. This equation is now in the standard quadratic form, where A = 2, B = -2, and C = -121. Rewriting the equation in this standard form is a crucial step as it sets the foundation for applying the completing the square method effectively. By ensuring the equation is in this format, we can systematically apply the algebraic manipulations required to find the solutions.

Removing the Leading Coefficient

The next crucial step in completing the square is to remove the leading coefficient, which is the coefficient of the x² term. In our standard form equation, 2x² - 2x - 121 = 0, the leading coefficient is 2. To remove this, we divide every term in the equation by 2. This process ensures that the coefficient of x² becomes 1, which is a requirement for the completing the square method to work effectively. Dividing each term by 2, we get: (2x²)/2 - (2x)/2 - 121/2 = 0/2. Simplifying this, we have: x² - x - 121/2 = 0. Now, the equation has a leading coefficient of 1, making it suitable for the next steps in completing the square. This process is essential because it simplifies the algebraic manipulations required in the subsequent steps. By dividing through by the leading coefficient, we avoid the complexities that can arise when dealing with a coefficient other than 1. This step is a critical part of the systematic approach to solving quadratic equations using the completing the square method. Ensuring that the leading coefficient is 1 makes the process more straightforward and less prone to errors.

Completing the Square

Now, we come to the heart of the method: completing the square. With the equation in the form x² - x - 121/2 = 0, we need to transform the left side into a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the form (x + a)² or (x - a)². To complete the square, we take half of the coefficient of the x term, square it, and add it to both sides of the equation. In our equation, the coefficient of the x term is -1. Half of -1 is -1/2, and squaring -1/2 gives us (-1/2)² = 1/4. So, we add 1/4 to both sides of the equation: x² - x + 1/4 - 121/2 = 1/4. Next, we move the constant term (-121/2) to the right side of the equation: x² - x + 1/4 = 1/4 + 121/2. Now, we simplify the right side of the equation. To add 1/4 and 121/2, we need a common denominator, which is 4. So, we rewrite 121/2 as 242/4. Thus, the right side becomes 1/4 + 242/4 = 243/4. Our equation is now: x² - x + 1/4 = 243/4. The left side of the equation is now a perfect square trinomial. It can be factored as (x - 1/2)². So, we rewrite the equation as (x - 1/2)² = 243/4. This step is the essence of the completing the square method. By adding the appropriate constant to both sides, we transform the quadratic expression into a form that is easily solvable.

Solving for x

With the equation in the form (x - 1/2)² = 243/4, we can now solve for x. The first step is to take the square root of both sides of the equation. This gives us: √(x - 1/2)² = ±√(243/4). Simplifying the left side, we get x - 1/2 = ±√(243/4). Now, we simplify the right side. The square root of 243 can be simplified as √(81 * 3) = 9√3, and the square root of 4 is 2. So, √(243/4) = (9√3)/2. Our equation becomes: x - 1/2 = ±(9√3)/2. To isolate x, we add 1/2 to both sides of the equation: x = 1/2 ± (9√3)/2. This gives us two possible solutions for x: x = (1 + 9√3)/2 and x = (1 - 9√3)/2. These are the roots of the original quadratic equation. Solving for x involves carefully applying algebraic operations to isolate the variable. Taking the square root introduces both positive and negative solutions, which is a critical aspect of solving quadratic equations. The final solutions represent the values of x that satisfy the original equation. This step demonstrates the power of the completing the square method in finding exact solutions, even when they involve radicals.

Verifying the Solutions

To ensure the accuracy of our solutions, it's crucial to verify them. We have two solutions for x: x = (1 + 9√3)/2 and x = (1 - 9√3)/2. To verify these solutions, we substitute each value back into the original equation, 2x² + 2x + 49 = 4x + 170, and check if both sides of the equation are equal. Let's start with the first solution, x = (1 + 9√3)/2. Substituting this value into the equation involves squaring the expression, multiplying by 2, adding 2 times the expression, and then adding 49. We then compare this result with 4 times the expression plus 170. This process can be computationally intensive but is necessary for verifying the solution. Similarly, we substitute the second solution, x = (1 - 9√3)/2, into the original equation and perform the same calculations. If both sides of the equation are equal for both solutions, we can confidently say that our solutions are correct. Verification is a critical step in the problem-solving process. It helps identify any errors made during the algebraic manipulations and ensures that the solutions obtained are accurate. This step reinforces the importance of precision and attention to detail in mathematical problem-solving.

Conclusion

In conclusion, the method of completing the square is a powerful technique for solving quadratic equations. By following the systematic steps of rewriting the equation in standard form, removing the leading coefficient, completing the square, solving for x, and verifying the solutions, we can accurately find the roots of any quadratic equation. Our example equation, 2x² + 2x + 49 = 4x + 170, presented by Edna Ann R. Cavalida, demonstrates the practical application of this method. This guide has provided a comprehensive explanation of each step, ensuring that students and anyone interested in algebra can understand and apply this technique effectively. Completing the square is not only a method for finding solutions but also a valuable tool for understanding the structure and properties of quadratic expressions. It reinforces algebraic skills and enhances problem-solving abilities. Mastering this method opens doors to more advanced topics in mathematics and provides a solid foundation for further studies. The ability to solve quadratic equations is essential in various fields, including physics, engineering, and economics. Therefore, understanding and mastering the completing the square method is a valuable asset for anyone pursuing these disciplines. By practicing and applying this method to a variety of problems, you can develop confidence and proficiency in solving quadratic equations.