Solving X^2 + 4x - 4 = 8 A Step-by-Step Guide
Solve for x in the equation x^2+4x-4=8
Introduction
In this article, we will delve into the process of solving for x in the quadratic equation x² + 4x - 4 = 8. Quadratic equations are a fundamental topic in algebra, and mastering their solutions is crucial for various mathematical and scientific applications. We will explore the different methods available to solve this equation, including factoring, completing the square, and using the quadratic formula. Each method offers a unique approach to finding the values of x that satisfy the equation. Understanding these methods not only helps in solving this specific problem but also equips you with the skills to tackle a wide range of quadratic equations. So, let's embark on this mathematical journey and unravel the solutions to this equation.
Understanding Quadratic Equations
Before diving into the solution, let's first understand what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree. This means that the highest power of the variable (in our 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 0. The solutions to a quadratic equation are also called roots or zeros of the equation. These are the values of x that make the equation true. Quadratic equations can have two real solutions, one real solution (a repeated root), or two complex solutions. The nature of the solutions depends on the discriminant, which is given by the formula b² - 4ac. If the discriminant is positive, there are two distinct real solutions; if it is zero, there is one real solution; and if it is negative, there are two complex solutions. In our given equation, x² + 4x - 4 = 8, we first need to rewrite it in the standard form by subtracting 8 from both sides, which gives us x² + 4x - 12 = 0. Now, we can identify the coefficients: a = 1, b = 4, and c = -12. This sets the stage for us to explore various methods for finding the values of x that satisfy this equation. Let's proceed to the methods of solving quadratic equations.
Method 1: Factoring
One of the most straightforward methods for solving quadratic equations is factoring. Factoring involves expressing the quadratic equation as a product of two binomials. This method is particularly effective when the equation can be easily factored. To factor the quadratic equation x² + 4x - 12 = 0, we need to find two numbers that multiply to -12 (the constant term) and add up to 4 (the coefficient of the x term). Let's think about the factors of -12. We have pairs like (1, -12), (-1, 12), (2, -6), (-2, 6), (3, -4), and (-3, 4). Among these pairs, the numbers -2 and 6 satisfy our conditions because (-2) * 6 = -12 and (-2) + 6 = 4. Therefore, we can rewrite the quadratic equation as (x - 2)(x + 6) = 0. Now, the equation is in factored form, which means that the product of the two binomials is zero. For this product to be zero, at least one of the binomials must be zero. This gives us two separate linear equations: x - 2 = 0 and x + 6 = 0. Solving the first equation, x - 2 = 0, we add 2 to both sides, which yields x = 2. Solving the second equation, x + 6 = 0, we subtract 6 from both sides, which yields x = -6. Thus, the solutions to the quadratic equation x² + 4x - 12 = 0 are x = 2 and x = -6. This method highlights the elegance of factoring in solving quadratic equations when the factors are readily apparent. However, not all quadratic equations can be easily factored, which leads us to explore other methods.
Method 2: Completing the Square
Another powerful method for solving quadratic equations is completing the square. This method transforms the quadratic equation into a perfect square trinomial, which can then be easily solved by taking the square root. To apply this method to the equation x² + 4x - 12 = 0, we first isolate the constant term by adding 12 to both sides, giving us x² + 4x = 12. Next, we need to complete the square on the left-hand side. To do this, we take half of the coefficient of the x term (which is 4), square it, and add it to both sides of the equation. Half of 4 is 2, and 2 squared is 4. So, we add 4 to both sides of the equation: x² + 4x + 4 = 12 + 4. This simplifies to x² + 4x + 4 = 16. Now, the left-hand side is a perfect square trinomial, which can be factored as (x + 2)². So, we have (x + 2)² = 16. To solve for x, we take the square root of both sides of the equation. Remember that when taking the square root, we need to consider both the positive and negative roots. So, √((x + 2)²) = ±√16, which gives us x + 2 = ±4. This leads to two separate equations: x + 2 = 4 and x + 2 = -4. Solving the first equation, x + 2 = 4, we subtract 2 from both sides, which yields x = 2. Solving the second equation, x + 2 = -4, we subtract 2 from both sides, which yields x = -6. Therefore, the solutions to the quadratic equation x² + 4x - 12 = 0, obtained by completing the square, are x = 2 and x = -6. Completing the square is a versatile method that can be used to solve any quadratic equation, even those that are not easily factored. It provides a systematic approach to finding the solutions and is particularly useful when deriving the quadratic formula.
Method 3: Quadratic Formula
The quadratic formula is a universal method for solving quadratic equations of the form ax² + bx + c = 0. It provides a direct way to find the solutions without the need for factoring or completing the square. The formula is given by: x = (-b ± √(b² - 4ac)) / (2a). To apply the quadratic formula to our equation, x² + 4x - 12 = 0, we first identify the coefficients: a = 1, b = 4, and c = -12. Now, we substitute these values into the quadratic formula: x = (-4 ± √(4² - 4 * 1 * -12)) / (2 * 1). Let's simplify the expression step by step. First, we calculate the discriminant, which is the term inside the square root: 4² - 4 * 1 * -12 = 16 + 48 = 64. So, the equation becomes: x = (-4 ± √64) / 2. The square root of 64 is 8, so we have: x = (-4 ± 8) / 2. This gives us two possible solutions: x = (-4 + 8) / 2 and x = (-4 - 8) / 2. For the first solution, x = (-4 + 8) / 2 = 4 / 2 = 2. For the second solution, x = (-4 - 8) / 2 = -12 / 2 = -6. Thus, the solutions to the quadratic equation x² + 4x - 12 = 0, using the quadratic formula, are x = 2 and x = -6. The quadratic formula is a powerful tool that guarantees a solution for any quadratic equation, regardless of its factorability. It is an essential method in algebra and is widely used in various mathematical and scientific fields.
Comparing the Methods
We have now explored three different methods for solving the quadratic equation x² + 4x - 4 = 8 (which we rewrote as x² + 4x - 12 = 0): factoring, completing the square, and the quadratic formula. Each method has its strengths and weaknesses, and the choice of method often depends on the specific equation and personal preference. Factoring is the most efficient method when the quadratic equation can be easily factored. It involves expressing the equation as a product of two binomials and setting each binomial equal to zero to find the solutions. However, not all quadratic equations are easily factorable, which limits the applicability of this method. Completing the square is a more versatile method that can be used to solve any quadratic equation. It involves transforming the equation into a perfect square trinomial and then taking the square root to find the solutions. While this method is more systematic than factoring, it can be more complex and time-consuming, especially when the coefficients are not integers. The quadratic formula is the most general method for solving quadratic equations. It provides a direct way to find the solutions by substituting the coefficients of the equation into a formula. This method is guaranteed to work for any quadratic equation, regardless of its factorability or the complexity of its coefficients. However, the quadratic formula can be computationally intensive, especially when the coefficients are large or involve radicals. In our example, all three methods led to the same solutions: x = 2 and x = -6. This demonstrates the consistency of these methods and reinforces the importance of understanding multiple approaches to solving quadratic equations. Ultimately, the best method to use depends on the specific problem and the solver's comfort level with each technique.
Conclusion
In conclusion, we have successfully solved the quadratic equation x² + 4x - 4 = 8 (rewritten as x² + 4x - 12 = 0) using three different methods: factoring, completing the square, and the quadratic formula. Each method provided a unique approach to finding the solutions, and all three methods yielded the same results: x = 2 and x = -6. This exercise highlights the importance of understanding various techniques for solving quadratic equations and reinforces the fundamental principles of algebra. Factoring is an efficient method for easily factorable equations, while completing the square offers a systematic approach for any quadratic equation. The quadratic formula provides a universal solution, guaranteeing results regardless of the equation's complexity. Mastering these methods equips you with the skills to tackle a wide range of mathematical problems and enhances your problem-solving abilities. Quadratic equations are a cornerstone of algebra, and a thorough understanding of their solutions is crucial for further studies in mathematics and related fields. By exploring these different methods, we have gained a deeper appreciation for the beauty and versatility of algebraic techniques. Remember to practice these methods with various quadratic equations to solidify your understanding and build confidence in your problem-solving skills.