Solving 5x² - 1 = 9 A Step-by-Step Mathematical Guide

Introduction: Unraveling Quadratic Equations

In the realm of mathematics, quadratic equations hold a significant place, serving as fundamental tools for modeling various real-world phenomena. These equations, characterized by a variable raised to the power of two, often present a unique challenge to solve. In this comprehensive guide, we will delve into the process of solving the specific quadratic equation 5x² - 1 = 9, providing a step-by-step approach that will not only lead you to the solution but also enhance your understanding of the underlying mathematical principles. Before we embark on the solution journey, it's crucial to grasp the core concepts of quadratic equations and the various methods available to tackle them. A quadratic equation is generally expressed in the form ax² + bx + c = 0, where a, b, and c are constants, and x represents the unknown variable. The solutions to this equation, also known as roots or zeros, are the values of x that satisfy the equation. These roots can be real or complex numbers, depending on the nature of the coefficients a, b, and c. Several methods exist for solving quadratic equations, each with its own strengths and weaknesses. Some of the most common techniques include factoring, completing the square, and employing the quadratic formula. Factoring involves expressing the quadratic expression as a product of two linear factors, which can then be solved individually to find the roots. Completing the square is a method that transforms the equation into a perfect square trinomial, allowing for easy extraction of the roots. The quadratic formula, a general solution derived from completing the square, provides a direct way to calculate the roots for any quadratic equation, regardless of its factorability. In the case of our equation, 5x² - 1 = 9, we will explore multiple approaches to find the solutions, showcasing the versatility of mathematical problem-solving. By the end of this guide, you will not only be equipped with the knowledge to solve this specific equation but also gain a deeper appreciation for the elegance and power of quadratic equations.

Method 1: Isolating the Variable and Using Square Roots

This method involves strategically manipulating the equation to isolate the term containing the variable (x²) on one side and then taking the square root of both sides to solve for x. It's a straightforward approach that works well when the equation can be easily rearranged in this manner. To begin, let's revisit our equation: 5x² - 1 = 9. Our primary goal is to isolate the x² term. The first step in this process is to eliminate the constant term (-1) on the left side of the equation. We can achieve this by adding 1 to both sides of the equation. This maintains the equality and moves us closer to our objective. Adding 1 to both sides, we get: 5x² - 1 + 1 = 9 + 1, which simplifies to 5x² = 10. Now that we have isolated the term with x², we need to get rid of the coefficient (5) multiplying it. To do this, we divide both sides of the equation by 5. This ensures that the equality remains intact while isolating x² completely. Dividing both sides by 5, we have: 5x² / 5 = 10 / 5, which simplifies to x² = 2. We've successfully isolated x². The next step is to undo the square. To do this, 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, as both will satisfy the equation. Taking the square root of both sides, we get: √(x²) = ±√2. This simplifies to x = ±√2. Therefore, the solutions to the equation 5x² - 1 = 9 are x = √2 and x = -√2. These are the exact solutions, expressed in terms of the square root of 2. If you need a decimal approximation, you can use a calculator to find that √2 is approximately 1.414. Thus, the approximate solutions are x ≈ 1.414 and x ≈ -1.414. This method of isolating the variable and using square roots is a powerful technique for solving quadratic equations that are in a specific form. It's efficient and provides a clear path to the solutions. However, it's not always applicable to all quadratic equations. In cases where the equation has a linear term (i.e., a term with x to the power of 1), other methods, such as factoring, completing the square, or the quadratic formula, may be more appropriate. In the following sections, we'll explore alternative methods to solve this equation, demonstrating the versatility of mathematical problem-solving.

Method 2: Using the Quadratic Formula

The quadratic formula is a universal tool for solving quadratic equations of the form ax² + bx + c = 0. It provides a direct way to calculate the roots, regardless of whether the equation can be factored or easily rearranged. This method is particularly useful when dealing with equations that have complex coefficients or do not factor neatly. The quadratic formula is derived by completing the square on the general quadratic equation ax² + bx + c = 0. It states that the solutions for x are given by:

x = (-b ± √(b² - 4ac)) / 2a

To apply the quadratic formula to our equation, 5x² - 1 = 9, we first need to rewrite it in the standard form ax² + bx + c = 0. Subtracting 9 from both sides, we get: 5x² - 1 - 9 = 0, which simplifies to 5x² - 10 = 0. Now we can identify the coefficients: a = 5, b = 0 (since there is no x term), and c = -10. Substituting these values into the quadratic formula, we get:

x = (-0 ± √(0² - 4 * 5 * -10)) / (2 * 5)

Simplifying the expression under the square root:

x = (± √(200)) / 10

We can simplify √200 by factoring out the largest perfect square: √200 = √(100 * 2) = √100 * √2 = 10√2. Substituting this back into the equation:

x = (± 10√2) / 10

Now, we can cancel the common factor of 10:

x = ±√2

Therefore, the solutions to the equation 5x² - 1 = 9 using the quadratic formula are x = √2 and x = -√2, which are the same solutions we obtained using the method of isolating the variable and taking square roots. The quadratic formula, while being a powerful and versatile tool, can sometimes involve more calculations than other methods, especially when the coefficients are simple. However, its advantage lies in its generality; it can be applied to any quadratic equation, making it an indispensable technique in mathematics. In this case, the quadratic formula confirms our previous results and demonstrates its reliability in solving quadratic equations. In the next section, we'll briefly discuss another method, completing the square, to further illustrate the different approaches to solving quadratic equations and their interconnections.

Method 3: Completing the Square (Brief Overview)

Completing the square is another method for solving quadratic equations. While it is less commonly used for simple equations like 5x² - 1 = 9 (as the previous methods are more efficient in this case), understanding it provides a deeper insight into the structure of quadratic equations and their solutions. The basic idea behind completing the square is to transform the quadratic expression into a perfect square trinomial, which can then be easily factored and solved. For the equation 5x² - 1 = 9, we can rewrite it as 5x² = 10 and then x² = 2. Since there is no x term, the expression x² is already a perfect square (with the missing terms being zero). Therefore, we can directly take the square root of both sides, as we did in Method 1, to obtain x = ±√2. While completing the square might seem like an overkill for this particular equation, it's a fundamental technique for deriving the quadratic formula and for solving more complex quadratic equations. It's also useful in various other mathematical contexts, such as finding the vertex of a parabola. In summary, completing the square involves manipulating the quadratic equation to create a perfect square trinomial on one side, allowing for easy solution. However, for our equation 5x² - 1 = 9, the methods of isolating the variable and using square roots or the quadratic formula are more efficient. This brief overview highlights the interconnectedness of different methods in mathematics and how understanding multiple approaches can enhance problem-solving skills.

Conclusion: Mastering Quadratic Equation Solutions

In this comprehensive guide, we have successfully navigated the process of solving the quadratic equation 5x² - 1 = 9. We explored three distinct methods: isolating the variable and using square roots, applying the quadratic formula, and briefly discussing completing the square. Each method offered a unique perspective and reinforced the fundamental principles of solving quadratic equations. The method of isolating the variable and using square roots proved to be the most straightforward approach for this particular equation, as it directly led to the solutions x = √2 and x = -√2. This method is particularly effective when the equation can be easily rearranged to isolate the squared term. The quadratic formula, a universal tool for solving quadratic equations, provided an alternative route to the same solutions. While it may involve more calculations in simpler cases, its power lies in its applicability to any quadratic equation, regardless of its complexity. By substituting the coefficients of our equation into the quadratic formula, we reaffirmed the solutions obtained through the first method, highlighting the consistency and reliability of mathematical techniques. Completing the square, although not the most efficient method for this specific equation, offered a valuable insight into the underlying structure of quadratic equations. Understanding this method is crucial for deriving the quadratic formula and for tackling more challenging problems. It emphasizes the importance of having a diverse toolkit of mathematical techniques. Throughout this guide, we have emphasized the importance of understanding the mathematical concepts behind the solutions. Simply memorizing formulas is not enough; a deep understanding allows for flexibility and adaptability in problem-solving. By exploring multiple methods, we have demonstrated that there is often more than one way to arrive at the correct answer. This encourages critical thinking and the development of problem-solving strategies. Furthermore, this exercise in solving 5x² - 1 = 9 serves as a stepping stone to more advanced mathematical concepts. Quadratic equations form the foundation for various topics, including calculus, complex numbers, and mathematical modeling. Mastering these fundamentals is essential for future success in mathematics and related fields. In conclusion, solving quadratic equations is a fundamental skill in mathematics. By understanding the different methods available and the underlying principles, you can confidently tackle a wide range of problems. The solutions to 5x² - 1 = 9, x = √2 and x = -√2, represent not just numerical answers, but a deeper understanding of mathematical problem-solving.