Solving 5x² - 5 = 0 A Step-by-Step Guide And Explanation

In the realm of mathematics, quadratic equations hold a significant position, serving as fundamental tools for modeling and solving a wide array of real-world problems. These equations, characterized by their polynomial form with a highest degree of two, find applications in diverse fields such as physics, engineering, economics, and computer science. This article delves into the intricacies of solving the quadratic equation 5x² - 5 = 0, providing a step-by-step guide and exploring the underlying mathematical principles. Understanding how to solve quadratic equations is a fundamental skill in algebra and is essential for various applications in mathematics and other fields.

At its core, a quadratic equation is an equation that can be expressed in the standard form of ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The term 'quadratic' originates from the Latin word 'quadratus,' meaning square, as the equation involves a term with the variable raised to the power of two. The solutions to a quadratic equation, also known as its roots or zeros, represent the values of the variable that satisfy the equation. These roots can be real or complex numbers, depending on the coefficients of the equation.

The quadratic equation 5x² - 5 = 0 exemplifies a specific type of quadratic equation where the coefficient 'b' is zero. This simplified form allows for a more direct approach to finding the solutions. In this article, we will explore the step-by-step method to solve this equation, highlighting the underlying algebraic principles and demonstrating how to arrive at the correct solutions. Solving quadratic equations like this not only reinforces algebraic skills but also lays the groundwork for tackling more complex mathematical problems.

The methods for solving quadratic equations include factoring, completing the square, and using the quadratic formula. Each method has its advantages and is suitable for different types of quadratic equations. In the case of 5x² - 5 = 0, we will primarily focus on algebraic manipulation to isolate the variable and find its values. This approach is particularly effective when the equation can be easily simplified. Before we delve into the solution, it is important to understand the properties of quadratic equations and how their structure influences the methods we use to solve them.

Solving the quadratic equation 5x² - 5 = 0 involves a series of algebraic steps aimed at isolating the variable 'x'. This process reveals the values of 'x' that satisfy the equation, which are the solutions or roots of the equation. Understanding each step is crucial for mastering the technique of solving quadratic equations. Let's break down the solution process into manageable steps:

1. Isolate the x² term: The initial step involves isolating the term containing x² on one side of the equation. This is achieved by adding 5 to both sides of the equation. The equation 5x² - 5 = 0 transforms into 5x² = 5. This manipulation ensures that we are one step closer to isolating 'x' and finding its possible values. Adding the same value to both sides maintains the equation's balance, a fundamental principle in algebra.

2. Divide by the coefficient of x²: Next, to further isolate x², we divide both sides of the equation by the coefficient of x², which is 5. This step simplifies the equation to x² = 1. Dividing both sides by the same non-zero number preserves the equality and moves us closer to solving for 'x'. This step is crucial because it sets the stage for the final operation to find 'x'.

3. Take the square root of both sides: To solve for 'x', we take the square root of both sides of the equation. The square root operation is the inverse of squaring, allowing us to undo the square on 'x'. When taking the square root, it's crucial to remember that there are two possible solutions: a positive and a negative root. Therefore, the equation x² = 1 yields two solutions: x = 1 and x = -1. The positive and negative roots both satisfy the original equation, as both 1² and (-1)² equal 1.

4. Verify the solutions: Finally, it's a good practice to verify the solutions by substituting them back into the original equation. For x = 1, we have 5(1)² - 5 = 5 - 5 = 0, which confirms that x = 1 is a valid solution. Similarly, for x = -1, we have 5(-1)² - 5 = 5 - 5 = 0, which also confirms that x = -1 is a valid solution. This verification step ensures that no errors were made during the solution process and that the solutions obtained are accurate.

By following these steps, we successfully solved the quadratic equation 5x² - 5 = 0 and found the solutions x = 1 and x = -1. This method highlights the importance of algebraic manipulation and the understanding of inverse operations in solving equations. The solutions represent the points where the quadratic function intersects the x-axis, further illustrating the connection between algebra and graphical representations.

While the algebraic method we used to solve 5x² - 5 = 0 is straightforward and efficient for this particular equation, it's important to recognize that other methods exist for solving quadratic equations. These alternative methods, such as factoring, completing the square, and using the quadratic formula, offer different approaches and are suitable for various types of quadratic equations. Understanding these methods broadens our problem-solving toolkit and allows us to tackle a wider range of mathematical challenges.

• Factoring: Factoring involves expressing the quadratic equation as a product of two binomials. This method is particularly effective when the quadratic equation can be easily factored. For instance, if we had the equation x² - 4 = 0, we could factor it as (x - 2)(x + 2) = 0, leading to the solutions x = 2 and x = -2. However, in the case of 5x² - 5 = 0, factoring is not as direct, but we can factor out the common factor of 5 to get 5(x² - 1) = 0, which further factors into 5(x - 1)(x + 1) = 0, yielding the same solutions x = 1 and x = -1. Factoring is a powerful technique that relies on recognizing patterns and applying algebraic identities.

• Completing the square: Completing the square is a method that transforms the quadratic equation into a perfect square trinomial. This method is useful when the equation cannot be easily factored. For the general quadratic equation ax² + bx + c = 0, completing the square involves adding and subtracting a constant term to make the expression a perfect square. While this method can be applied to any quadratic equation, it is generally more complex than factoring or using the quadratic formula for simpler equations like 5x² - 5 = 0. Completing the square provides a systematic way to solve quadratic equations, but it requires a solid understanding of algebraic manipulation.

• Quadratic Formula: The quadratic formula is a universal method for solving quadratic equations of the form ax² + bx + c = 0. The formula is given by x = (-b ± √(b² - 4ac)) / (2a). This formula can be applied to any quadratic equation, regardless of its factorability. For the equation 5x² - 5 = 0, we have a = 5, b = 0, and c = -5. Plugging these values into the quadratic formula, we get x = (0 ± √(0² - 4(5)(-5))) / (2(5)), which simplifies to x = (± √100) / 10, leading to x = ± 1. The quadratic formula is a reliable tool, especially for equations that are difficult to solve using other methods. It ensures that we can find the solutions, whether they are real or complex.

Each of these methods offers a unique approach to solving quadratic equations. The choice of method often depends on the specific equation and personal preference. For 5x² - 5 = 0, the algebraic method we initially used is the most efficient. However, understanding factoring, completing the square, and the quadratic formula provides a comprehensive understanding of quadratic equations and their solutions. These alternative methods are not just tools for solving equations; they also deepen our understanding of algebraic principles and mathematical relationships.

Quadratic equations are not just abstract mathematical concepts; they have numerous real-world applications across various fields. Understanding these applications helps us appreciate the practical significance of quadratic equations and their role in solving real-world problems. From physics to engineering to economics, quadratic equations provide a powerful framework for modeling and analyzing various phenomena.

• Physics: In physics, quadratic equations are used to describe projectile motion, which is the motion of an object thrown or launched into the air. The trajectory of a projectile, such as a ball thrown upwards, can be modeled using a quadratic equation. The equation takes into account the initial velocity, launch angle, and gravitational acceleration. By solving the quadratic equation, we can determine the maximum height reached by the projectile, the time it takes to reach that height, and the total time the projectile is in the air. Similarly, quadratic equations are used in the study of optics to describe the focusing of light by lenses and mirrors. The lens equation, which relates the object distance, image distance, and focal length of a lens, is a quadratic equation. Solving this equation allows us to determine the position and size of images formed by optical systems.

• Engineering: Engineers use quadratic equations in various applications, including structural design and electrical circuits. In structural design, quadratic equations are used to calculate the forces and stresses on beams and other structural elements. For example, the deflection of a beam under load can be modeled using a quadratic equation. By solving this equation, engineers can ensure that the structure can withstand the applied loads without failing. In electrical circuits, quadratic equations are used to analyze circuits containing resistors, capacitors, and inductors. The behavior of these circuits can be described using differential equations, which often lead to quadratic equations. Solving these equations allows engineers to determine the current and voltage in the circuit as a function of time.

• Economics: Quadratic equations are used in economics to model various relationships, such as cost, revenue, and profit. For example, the cost of producing a certain number of goods can be modeled using a quadratic equation, where the cost increases as the square of the quantity produced. Similarly, the revenue from selling goods can be modeled using a quadratic equation, where the revenue depends on the price and quantity sold. By analyzing these quadratic equations, economists can determine the optimal production level and pricing strategy to maximize profit. Quadratic equations also play a role in financial modeling, such as in calculating compound interest and the time value of money.

• Computer Science: In computer science, quadratic equations are used in various algorithms and optimization problems. For example, quadratic programming is a type of optimization problem where the objective function is a quadratic equation and the constraints are linear equations. This technique is used in machine learning, portfolio optimization, and other applications. Quadratic equations are also used in computer graphics to model curves and surfaces. Bezier curves and B-spline curves, which are commonly used in computer-aided design (CAD) and animation, are defined using quadratic or higher-degree polynomials. The solutions to these equations determine the shape and properties of the curves.

The applications of quadratic equations extend beyond these examples, demonstrating their versatility and importance in various disciplines. By understanding the principles of quadratic equations and their solutions, we gain a powerful tool for analyzing and solving real-world problems. The ability to apply these concepts is essential for professionals in various fields and highlights the practical value of mathematical knowledge.

In conclusion, mastering quadratic equations is a crucial step in mathematical education, providing a foundation for more advanced topics and offering practical tools for solving real-world problems. This article has provided a comprehensive guide to solving the quadratic equation 5x² - 5 = 0, illustrating the step-by-step process and exploring alternative methods. By understanding the algebraic manipulations involved, we can confidently tackle similar equations and apply these techniques in various contexts. The journey through solving quadratic equations is not just about finding solutions; it's about developing critical thinking skills and problem-solving abilities.

We began by understanding the basics of quadratic equations, recognizing their standard form ax² + bx + c = 0 and the importance of identifying the coefficients a, b, and c. We then delved into the specific equation 5x² - 5 = 0, demonstrating how to isolate the x² term, divide by the coefficient, and take the square root of both sides to find the solutions x = 1 and x = -1. This method, while straightforward, highlights the fundamental principles of algebraic manipulation and the significance of inverse operations. The process of solving quadratic equations reinforces the concept that mathematical problems can be broken down into manageable steps, each building upon the previous one.

Furthermore, we explored alternative methods for solving quadratic equations, including factoring, completing the square, and using the quadratic formula. Each method offers a unique approach and is suitable for different types of equations. Factoring is efficient when the equation can be easily expressed as a product of binomials, while completing the square provides a systematic way to transform the equation into a perfect square trinomial. The quadratic formula, a universal method, guarantees a solution for any quadratic equation, regardless of its factorability. Understanding these different methods enhances our problem-solving versatility and allows us to choose the most appropriate technique for a given equation. This broader perspective is essential for tackling more complex mathematical problems in the future.

The real-world applications of quadratic equations, ranging from physics and engineering to economics and computer science, underscore their practical significance. In physics, quadratic equations model projectile motion and optics; in engineering, they are used in structural design and electrical circuits; in economics, they help analyze cost, revenue, and profit; and in computer science, they play a role in algorithms and computer graphics. These diverse applications demonstrate the power of quadratic equations as a tool for modeling and analyzing real-world phenomena. Recognizing these applications motivates us to further explore mathematical concepts and their relevance in various fields.

In conclusion, the ability to solve quadratic equations is not just a mathematical skill; it's a life skill. It enhances our logical reasoning, problem-solving abilities, and analytical thinking. By mastering quadratic equations, we gain confidence in our mathematical abilities and open doors to a deeper understanding of the world around us. The journey of learning mathematics is a continuous process, and quadratic equations serve as a crucial milestone in this journey. As we continue to explore more advanced topics, the fundamental principles learned here will serve as a solid foundation for future success.