Solving X(x-2) = 48 A Step-by-Step Guide

Jul 12, 2025 by ADMIN 41 views

Solving the Quadratic Equation x(x-2) = 48: A Comprehensive Guide

In this article, we will delve into the process of solving the quadratic equation x(x-2) = 48. This equation, while seemingly simple, provides a great opportunity to illustrate several fundamental concepts in algebra, including the distributive property, the standard form of a quadratic equation, and the quadratic formula. Our journey will start with simplifying the equation, transforming it into its standard form, and then exploring different methods to find the solutions, also known as roots, of the equation. We will emphasize the importance of understanding each step, ensuring that you not only get the correct answers but also grasp the underlying principles. This article is designed for students and enthusiasts of mathematics who wish to deepen their understanding of quadratic equations and problem-solving techniques. By the end of this guide, you will be well-equipped to tackle similar problems and appreciate the elegance and power of algebraic methods.

Before we dive into the specifics of solving x(x-2) = 48, it is crucial to understand what quadratic equations are and why they are important. 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 ≠ 0. Quadratic equations appear in various fields, from physics and engineering to economics and computer science. They are used to model parabolic trajectories, calculate areas, and even optimize processes. The solutions to a quadratic equation, also known as roots or zeros, represent the values of x that make the equation true. A quadratic equation can have two distinct real roots, one real root (a repeated root), or two complex roots. The nature of the roots depends on the discriminant, which is given by the formula Δ = b² - 4ac. If Δ > 0, the equation has two distinct real roots. If Δ = 0, the equation has one real root. And if Δ < 0, the equation has two complex roots. Understanding these fundamental concepts is essential for solving quadratic equations efficiently and accurately. In the following sections, we will apply these concepts to solve the equation x(x-2) = 48, demonstrating practical techniques and problem-solving strategies. By mastering quadratic equations, you unlock a powerful tool for tackling a wide range of mathematical and real-world problems.

Step 1: Expanding and Rearranging the Equation

The first step in solving the equation x(x-2) = 48 is to expand the left side using the distributive property. The distributive property states that a(b + c) = ab + ac. Applying this property to our equation, we get:

x(x - 2) = x * x - x * 2 = x² - 2x

Now our equation looks like this:

x² - 2x = 48

To solve a quadratic equation, it's essential to bring it into the standard form, which is ax² + bx + c = 0. To achieve this, we need to move the constant term (48) from the right side to the left side. We can do this by subtracting 48 from both sides of the equation:

x² - 2x - 48 = 48 - 48

x² - 2x - 48 = 0

Now the equation is in the standard quadratic form, where a = 1, b = -2, and c = -48. This form is crucial for applying various methods to solve the equation, such as factoring, completing the square, or using the quadratic formula. Each of these methods has its own advantages and is suitable for different types of quadratic equations. Factoring, for instance, is efficient when the quadratic expression can be easily factored into two binomials. Completing the square is a more general method that can be used for any quadratic equation. And the quadratic formula is a universal method that always provides the solutions, regardless of the complexity of the equation. In the next steps, we will explore these methods and determine the most appropriate one for solving our equation. By understanding the importance of the standard form and the different solution methods, you will be better equipped to tackle a wide variety of quadratic equation problems.

Step 2: Solving by Factoring

One of the most efficient methods to solve a quadratic equation, when applicable, is factoring. Factoring involves breaking down the quadratic expression into the product of two binomials. This method relies on the principle that if the product of two factors is zero, then at least one of the factors must be zero. In our case, we have the equation x² - 2x - 48 = 0. To factor this quadratic expression, we need to find two numbers that multiply to -48 (the constant term) and add up to -2 (the coefficient of the x term). Let's consider the factors of -48:

1 and -48
-1 and 48
2 and -24
-2 and 24
3 and -16
-3 and 16
4 and -12
-4 and 12
6 and -8
-6 and 8

Among these pairs, the numbers -6 and 8 satisfy our conditions: (-6) * 8 = -48 and (-6) + 8 = 2. However, we need the numbers to add up to -2, not 2. Therefore, we should consider the pair 6 and -8, where 6 * (-8) = -48 and 6 + (-8) = -2. Now we can rewrite the quadratic expression using these numbers:

x² - 2x - 48 = (x + 6)(x - 8)

So, our equation becomes:

(x + 6)(x - 8) = 0

Now, we can apply the zero-product property, which states that if ab = 0, then either a = 0 or b = 0 (or both). Applying this property to our equation, we set each factor equal to zero:

x + 6 = 0 or x - 8 = 0

Solving these linear equations, we get:

x = -6 or x = 8

Thus, the solutions to the quadratic equation x² - 2x - 48 = 0 are x = -6 and x = 8. Factoring is a powerful technique that simplifies the process of solving quadratic equations, especially when the roots are integers. In the next section, we will explore another method, the quadratic formula, which can be used to solve any quadratic equation, regardless of whether it can be easily factored.

Step 3: Verification of the Solutions

To ensure the accuracy of our solutions, it is crucial to verify them by substituting them back into the original equation. This step helps us catch any potential errors made during the solving process and provides confidence in our answers. We found that the solutions to the equation x(x-2) = 48 are x = -6 and x = 8. Let's substitute each of these values into the original equation and check if they hold true.

Verification for x = -6:

Substitute x = -6 into the original equation x(x-2) = 48:

(-6)((-6) - 2) = 48

(-6)(-8) = 48

48 = 48

Since the equation holds true for x = -6, it is a valid solution.

Verification for x = 8:

Now, substitute x = 8 into the original equation x(x-2) = 48:

(8)((8) - 2) = 48

(8)(6) = 48

48 = 48

Since the equation also holds true for x = 8, it is also a valid solution.

Both solutions, x = -6 and x = 8, satisfy the original equation. This verification step confirms that our solutions are correct and that we have successfully solved the quadratic equation. Verification is an essential part of problem-solving in mathematics, and it is always a good practice to check your answers, especially in exams and assessments. In the next section, we will summarize our findings and highlight the key steps involved in solving quadratic equations.

In this comprehensive guide, we have successfully solved the quadratic equation x(x-2) = 48. We began by understanding the basics of quadratic equations and their importance in various fields. We then transformed the given equation into the standard form ax² + bx + c = 0, which is essential for applying different solution methods. We employed the factoring method, a highly efficient technique when applicable, to find the solutions x = -6 and x = 8. Finally, we verified these solutions by substituting them back into the original equation, confirming their accuracy. This process highlights the importance of each step in solving quadratic equations, from simplifying and rearranging the equation to choosing the appropriate method and verifying the results. By mastering these techniques, you can confidently tackle a wide range of quadratic equation problems. Remember, practice is key to developing your problem-solving skills in mathematics. Keep exploring different types of equations and methods, and you will gradually build a strong foundation in algebra. The ability to solve quadratic equations is a valuable skill that will serve you well in various academic and real-world contexts. We hope this guide has been helpful in enhancing your understanding of quadratic equations and problem-solving strategies. Keep learning and exploring the fascinating world of mathematics!