In this article, we will delve into the process of solving the quadratic equation x² = -8x - 7 by employing the method of completing the square. This technique is a powerful tool for rewriting quadratic equations into a form that allows us to easily identify the solutions. We will break down the problem into two main steps. First, we will rewrite the given equation in the form (x + c)² = d or (x - c)² = d. Second, we will determine the solutions to the equation. Understanding the method of completing the square is crucial in algebra, as it provides a systematic way to solve quadratic equations, especially when factoring is not straightforward. This method not only helps in finding the roots of the equation but also provides a deeper understanding of the structure of quadratic expressions. The ability to manipulate quadratic equations and solve them efficiently is a fundamental skill in mathematics, with applications spanning various fields such as physics, engineering, and computer science. By the end of this article, you will have a comprehensive understanding of how to complete the square and use it to solve quadratic equations.
1. Rewriting the Equation by Completing the Square
To rewrite the equation x² = -8x - 7 by completing the square, our initial step is to bring all the terms to one side of the equation, leaving zero on the other side. This standard form of a quadratic equation is essential for applying the completing the square method. By rearranging the terms, we transform the equation into a more manageable format, which allows us to focus on manipulating the left-hand side to create a perfect square trinomial. This process involves adding and subtracting a specific value that makes the expression factorable into the form of a squared binomial. The significance of this step cannot be overstated, as it lays the foundation for the subsequent steps in the completing the square method. Without a clear and organized equation, the process of completing the square becomes significantly more challenging. The ability to manipulate equations and rearrange terms is a fundamental skill in algebra, and this initial step reinforces this concept. Furthermore, it highlights the importance of paying attention to detail and ensuring that all terms are correctly accounted for when working with algebraic expressions. Now, let's add 8x and 7 to both sides of the equation:
x² + 8x + 7 = 0
Now, we focus on the left-hand side of the equation, x² + 8x + 7. The core idea behind completing the square is to transform the quadratic expression into a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the square of a binomial, such as (x + a)² or (x - a)². To achieve this, we need to determine the constant term that, when added to the expression x² + 8x, will create a perfect square trinomial. The process involves taking half of the coefficient of the x term, squaring it, and then adding it to both sides of the equation. This step is crucial because it maintains the balance of the equation while allowing us to rewrite the expression in a more convenient form. The ability to identify and create perfect square trinomials is a key skill in algebra, and this step provides a practical application of this concept. By understanding the relationship between the coefficients of a quadratic expression and the resulting perfect square trinomial, students gain a deeper appreciation for the structure of algebraic expressions. The process also reinforces the importance of careful calculation and attention to detail, as errors in this step can lead to incorrect solutions. To find this constant, we take half of the coefficient of the x term, which is 8, and square it:
(8 / 2)² = 4² = 16
We add 16 to both sides of the equation. Adding the same value to both sides ensures that the equation remains balanced. This is a fundamental principle in algebra, as it allows us to manipulate equations without changing their solutions. The addition of 16 to both sides of the equation is a strategic move, as it allows us to rewrite the left-hand side as a perfect square trinomial. This transformation is the essence of the completing the square method, and it is crucial for solving quadratic equations. The ability to maintain the balance of an equation while performing algebraic manipulations is a key skill in mathematics, and this step reinforces this concept. Furthermore, it highlights the importance of understanding the properties of equality and how they can be used to solve problems. By carefully adding the correct value to both sides of the equation, we pave the way for the next step, which involves factoring the perfect square trinomial and simplifying the equation. So:
x² + 8x + 16 = -7 + 16
Now, we can rewrite the left-hand side as a squared binomial and simplify the right-hand side. This step is the culmination of the completing the square process, as it allows us to express the quadratic equation in a form that is easily solvable. The ability to factor a perfect square trinomial into a squared binomial is a fundamental skill in algebra, and this step provides a practical application of this concept. By recognizing the pattern of a perfect square trinomial, students can quickly and efficiently rewrite the expression in a more convenient form. The simplification of the right-hand side is equally important, as it reduces the equation to its simplest form, making it easier to solve. This step reinforces the importance of algebraic manipulation and simplification, which are essential skills in mathematics. Furthermore, it highlights the connection between different algebraic concepts and how they can be used together to solve problems. By successfully rewriting the equation in this form, we set the stage for the final step, which involves taking the square root of both sides and solving for x.
(x + 4)² = 9
2. Determining the Solutions to the Equation
Now that we have rewritten the equation in the form (x + 4)² = 9, we can proceed to find the solutions. The next step involves taking the square root of both sides of the equation. When taking the square root, it is crucial to remember that there are two possible solutions: a positive square root and a negative square root. This is because both the positive and negative values, when squared, will result in the same positive number. For example, both 3² and (-3)² equal 9. Therefore, when solving equations involving square roots, we must consider both possibilities to ensure that we find all possible solutions. The inclusion of both positive and negative square roots is a common source of error for students, so it is important to emphasize this step. By carefully considering both possibilities, we can avoid missing solutions and ensure that we have a complete understanding of the problem. This step reinforces the importance of attention to detail and careful consideration of all possible scenarios when solving mathematical problems. So, taking the square root of both sides gives us:
√(x + 4)² = ±√9
This simplifies to:
x + 4 = ±3
Now, we solve for x by considering both the positive and negative cases. This step involves isolating x on one side of the equation, which is a fundamental skill in algebra. By considering both the positive and negative cases, we ensure that we find all possible solutions to the equation. This step reinforces the importance of careful calculation and attention to detail, as errors in this step can lead to incorrect solutions. The ability to solve for x in different algebraic contexts is a key skill in mathematics, and this step provides a practical application of this concept. Furthermore, it highlights the connection between different algebraic concepts and how they can be used together to solve problems. By successfully solving for x in both cases, we arrive at the solutions to the original quadratic equation.
Case 1: x + 4 = 3
Subtracting 4 from both sides:
x = 3 - 4
x = -1
Case 2: x + 4 = -3
Subtracting 4 from both sides:
x = -3 - 4
x = -7
Thus, the solutions to the equation are x = -1 and x = -7.
In conclusion, we have successfully solved the quadratic equation x² = -8x - 7 by completing the square. This method involves rewriting the equation in the form (x + c)² = d, which we achieved by adding 16 to both sides of the equation after moving all terms to one side. We then factored the perfect square trinomial and took the square root of both sides, remembering to consider both positive and negative roots. This led us to the two solutions, x = -1 and x = -7. The process of completing the square is a versatile technique that can be applied to any quadratic equation, making it a valuable tool in algebra. It provides a systematic approach to solving equations, especially when factoring is not immediately obvious. Furthermore, it enhances our understanding of the structure of quadratic expressions and their properties. The ability to manipulate and solve quadratic equations is essential in various fields, including mathematics, physics, and engineering. By mastering the method of completing the square, students gain a deeper appreciation for algebraic concepts and develop problem-solving skills that are applicable in a wide range of contexts. The solutions we found, x = -1 and x = -7, represent the points where the parabola described by the equation intersects the x-axis. These values are crucial in understanding the behavior of the quadratic function and its applications in real-world scenarios. Therefore, the ability to accurately solve quadratic equations is a fundamental skill in mathematics, and the method of completing the square provides a reliable and effective approach.
