Solving Quadratic Equations A Step-by-Step Guide For 9x² - 5x = 4
Quadratic equations are a cornerstone of algebra, appearing in various fields ranging from physics and engineering to economics and computer science. The ability to solve them efficiently and accurately is a fundamental skill. In this comprehensive guide, we will delve into the step-by-step process of solving the quadratic equation 9x² - 5x = 4. We will explore different methods, including factoring, completing the square, and the quadratic formula, ultimately providing you with a thorough understanding of how to find the solutions to this equation. This knowledge will not only help you solve this specific problem but also equip you with the tools to tackle a wide range of quadratic equations. Mastering these techniques will significantly enhance your problem-solving abilities in mathematics and related disciplines. Throughout this guide, we will emphasize clarity and precision, ensuring that each step is explained in detail and the underlying concepts are thoroughly understood. By the end, you will confidently navigate quadratic equations and apply the appropriate methods to find their solutions. Let's embark on this journey to unlock the secrets of quadratic equations and sharpen your mathematical prowess.
H2: Understanding Quadratic Equations
Before we dive into solving the specific equation 9x² - 5x = 4, it's crucial to understand the basics of quadratic equations. A quadratic equation is a polynomial equation of the second degree, generally represented in the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The solutions to a quadratic equation are also known as its roots or zeros, which are the values of x that satisfy the equation. Quadratic equations can have two distinct real solutions, one real solution (a repeated root), or two complex solutions, depending on the discriminant (b² - 4ac). The discriminant plays a crucial role in determining the nature of the roots; if it's positive, there are two distinct real roots; if it's zero, there is one real root (repeated); and if it's negative, there are two complex roots. Understanding these fundamental concepts is essential for approaching any quadratic equation with confidence. We will revisit the discriminant later when we use the quadratic formula. By grasping the general form and the significance of the coefficients, you can better understand the methods used to solve these equations. This foundational knowledge will empower you to analyze quadratic equations more effectively and choose the most appropriate solution technique. Now that we have a solid understanding of what quadratic equations are, let's move on to the specific equation at hand and explore various methods to solve it.
H2: Transforming the Equation
The first step in solving 9x² - 5x = 4 is to rewrite the equation in the standard quadratic form, ax² + bx + c = 0. To do this, we subtract 4 from both sides of the equation: 9x² - 5x - 4 = 0. Now, the equation is in the standard form, where a = 9, b = -5, and c = -4. This form is essential for applying various solution methods, such as factoring, completing the square, or using the quadratic formula. Transforming the equation into this standard form allows us to clearly identify the coefficients, which are crucial for subsequent steps. It also provides a structured framework for solving the equation, ensuring that we don't miss any critical steps. By arranging the equation in this manner, we can systematically analyze the equation and select the most efficient method for finding the solutions. The standard form also facilitates the application of the quadratic formula, which is a general solution that can be used for any quadratic equation. Therefore, this initial transformation is a crucial prerequisite for solving the equation and sets the stage for the following steps. With the equation now in the standard form, we can proceed to explore different methods for finding the values of x that satisfy it. Let's move on to the next section and discuss one of the common methods: factoring.
H2: Solving by Factoring
Factoring is a powerful method for solving quadratic equations when the equation can be easily factored. To solve 9x² - 5x - 4 = 0 by factoring, we need to find two binomials that, when multiplied, give us the quadratic expression. This involves finding two numbers that multiply to ac (9 * -4 = -36) and add up to b (-5). After some consideration, we find that the numbers -9 and 4 satisfy these conditions (-9 * 4 = -36 and -9 + 4 = -5). Now, we rewrite the middle term (-5x) using these numbers: 9x² - 9x + 4x - 4 = 0. Next, we factor by grouping. From the first two terms, we can factor out 9x, and from the last two terms, we can factor out 4: 9x(x - 1) + 4(x - 1) = 0. Notice that we now have a common factor of (x - 1). Factoring this out, we get: (9x + 4)(x - 1) = 0. According to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we set each factor equal to zero and solve for x: 9x + 4 = 0 or x - 1 = 0. Solving 9x + 4 = 0 gives us x = -4/9, and solving x - 1 = 0 gives us x = 1. Thus, the solutions to the quadratic equation 9x² - 5x - 4 = 0 are x = -4/9 and x = 1. Factoring is often the quickest method when it's applicable, but it's not always straightforward. If factoring is difficult or not possible, we can turn to other methods, such as the quadratic formula, which we will discuss later. For this particular equation, factoring provided a clear and efficient path to the solutions. Let's now examine another method, the quadratic formula, to further solidify our understanding and problem-solving skills.
H2: Applying the 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, regardless of whether the equation can be factored easily. The quadratic formula is given by: x = [-b ± √(b² - 4ac)] / (2a). For the equation 9x² - 5x - 4 = 0, we have a = 9, b = -5, and c = -4. Plugging these values into the quadratic formula, we get: x = [-(-5) ± √((-5)² - 4 * 9 * -4)] / (2 * 9). Simplifying, we have: x = [5 ± √(25 + 144)] / 18. Further simplification gives us: x = [5 ± √169] / 18. Since √169 = 13, we have: x = [5 ± 13] / 18. Now, we have two possible solutions: x = (5 + 13) / 18 and x = (5 - 13) / 18. For the first solution: x = 18 / 18 = 1. For the second solution: x = -8 / 18 = -4/9. Thus, the solutions to the quadratic equation 9x² - 5x - 4 = 0 are x = 1 and x = -4/9, which match the solutions we found by factoring. The quadratic formula is a powerful tool that ensures we can always find the solutions to any quadratic equation. It is particularly useful when factoring is not readily apparent or when the solutions are not rational numbers. By understanding and applying the quadratic formula, you can confidently tackle any quadratic equation and find its solutions accurately. Now that we have successfully solved the equation using two different methods, let's summarize our findings and emphasize the importance of choosing the appropriate method based on the equation's characteristics.
H2: Solutions and Summary
In summary, we have solved the quadratic equation 9x² - 5x = 4 using two distinct methods: factoring and the quadratic formula. Both methods yielded the same solutions: x = 1 and x = -4/9. This demonstrates the versatility of quadratic equations and the different approaches we can take to solve them. Factoring is an efficient method when the quadratic expression can be easily factored, as it often provides a quicker path to the solutions. However, when factoring is not straightforward or when the equation involves coefficients that make factoring difficult, the quadratic formula provides a reliable and universal solution. The choice of method often depends on the specific characteristics of the equation and the solver's familiarity with each technique. Understanding both methods and knowing when to apply each one is crucial for effective problem-solving in algebra. The quadratic formula, in particular, is an invaluable tool that ensures we can always find the solutions, regardless of the equation's complexity. By mastering these methods, you will be well-equipped to handle a wide range of quadratic equations and apply them in various mathematical and real-world contexts. The ability to solve quadratic equations is a fundamental skill that underpins many areas of mathematics and science, making it an essential topic to understand thoroughly. With the solutions in hand and a clear understanding of the methods used, we can confidently say that we have successfully tackled this quadratic equation. Let's continue to practice and refine our skills in solving quadratic equations to further enhance our mathematical abilities.
x = 1, -4/9
