Transforming (5x-2)/2 - (19x+6)/(2x) = (3x-2)/4 Into General Quadratic Form

JU07/11/2025 08, 2025 by THE IDEN 76 views

Transforming a Rational Equation into General Quadratic Form

In the realm of algebra, quadratic equations hold a fundamental position. They appear in diverse applications, from physics and engineering to economics and computer science. A quadratic equation is generally expressed in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. Before solving a problem like this, it's important to grasp the concepts and methods required for solving quadratic equations. These equations frequently arise in diverse fields and mathematical situations. This discussion will delve into the process of transforming a given rational equation into the standard quadratic form, which is a crucial skill in algebra. The given rational equation is: (5x - 2)/2 - (19x + 6)/(2x) = (3x - 2)/4. Our aim is to manipulate this equation algebraically to fit the ax² + bx + c = 0 format. This involves eliminating fractions, combining like terms, and rearranging the equation. Understanding how to perform these transformations is essential for solving a wide range of algebraic problems. By converting the equation into general quadratic form, we can easily apply methods such as factoring, completing the square, or using the quadratic formula to find the solutions for x. This article will walk you through each step of the transformation process, providing a clear and concise explanation for students and math enthusiasts alike. Throughout this discussion, we will emphasize the importance of each algebraic step and its impact on the overall goal of obtaining the general quadratic form. This detailed approach will not only help in solving the current problem but also build a solid foundation for tackling more complex algebraic challenges in the future. Ultimately, mastering this skill enhances one's ability to approach and solve mathematical problems systematically and efficiently.

Step-by-Step Transformation

To convert the given rational equation into general quadratic form, we'll follow a series of algebraic steps. The equation we are starting with is (5x - 2)/2 - (19x + 6)/(2x) = (3x - 2)/4. The first crucial step in transforming this rational equation into the general quadratic form involves eliminating the fractions. This is accomplished by identifying the least common denominator (LCD) of all the fractions present in the equation. The denominators in our equation are 2, 2x, and 4. The least common denominator (LCD) for these terms is 4x. Multiplying each term of the equation by 4x will clear the fractions. Let's break this down further. When we multiply the first term, (5x - 2)/2, by 4x, we get 2x(5x - 2). For the second term, (19x + 6)/(2x), multiplying by 4x gives us 2(19x + 6). Lastly, multiplying the right-hand side, (3x - 2)/4, by 4x yields x(3x - 2). Now, the equation looks like this: 2x(5x - 2) - 2(19x + 6) = x(3x - 2). This step is critical because it simplifies the equation significantly, making it easier to manipulate further. By eliminating the fractions, we pave the way for subsequent steps such as expanding the terms and combining like terms. This process of finding and multiplying by the LCD is a fundamental technique in algebra, especially when dealing with rational expressions and equations. It ensures that we maintain the equality of the equation while transforming it into a more manageable form. The next step involves expanding each term. Distribute the terms outside the parentheses into the terms inside. This means multiplying 2x by both 5x and -2, multiplying -2 by both 19x and 6, and multiplying x by both 3x and -2. When we expand 2x(5x - 2), we get 10x² - 4x. Expanding -2(19x + 6) yields -38x - 12. And expanding x(3x - 2) gives us 3x² - 2x. So now our equation looks like this: 10x² - 4x - 38x - 12 = 3x² - 2x. The act of expanding the terms is vital because it eliminates the parentheses, allowing us to combine like terms in the subsequent steps. This process transforms the equation from a factored form to a more extended form, which is necessary for identifying and grouping similar terms. Careful attention to the signs and coefficients during expansion is crucial to avoid errors. This step sets the stage for simplifying the equation further and moving closer to the desired general quadratic form.

Combining Like Terms

After expanding the terms, the next step is to combine like terms on each side of the equation. This involves identifying terms with the same variable and exponent and then adding or subtracting their coefficients. Looking at our equation, 10x² - 4x - 38x - 12 = 3x² - 2x, we can see that on the left side, we have two terms with x: -4x and -38x. Combining these gives us -42x. The 10x² term and the constant term -12 remain as they are, since there are no other like terms to combine with them on the left side. On the right side, we have 3x² and -2x, which are already in their simplest form. So, after combining like terms, the equation becomes: 10x² - 42x - 12 = 3x² - 2x. This step is crucial for simplifying the equation and making it easier to rearrange into the standard quadratic form. By grouping and combining like terms, we reduce the number of terms in the equation, which aids in the subsequent steps of rearranging the equation. This process also helps in identifying the coefficients of each term, which will be important when we express the equation in the general quadratic form ax² + bx + c = 0. The ability to accurately combine like terms is a fundamental skill in algebra, and it is essential for solving various types of equations.

Rearranging into General Quadratic Form

Now that we have simplified the equation by combining like terms, the next crucial step is to rearrange it into the general quadratic form, which is ax² + bx + c = 0. Our current equation is 10x² - 42x - 12 = 3x² - 2x. To get it into the desired form, we need to move all terms to one side of the equation, leaving zero on the other side. Typically, it's preferable to move terms in such a way that the coefficient of x² remains positive, which can simplify subsequent steps like factoring or using the quadratic formula. To do this, we can subtract 3x² from both sides, add 2x to both sides, and leave the constant term on the left. Subtracting 3x² from both sides gives us 7x² - 42x - 12 = -2x. Adding 2x to both sides yields 7x² - 40x - 12 = 0. Now the equation is in the general quadratic form. In this form, we can easily identify the coefficients a, b, and c. Here, a = 7, b = -40, and c = -12. Rearranging the equation into this standard form is vital because it allows us to directly apply various methods for solving quadratic equations, such as factoring, completing the square, or using the quadratic formula. The general quadratic form provides a clear and structured representation of the equation, which is essential for these solution methods. Furthermore, this form is universally recognized and facilitates the comparison and analysis of different quadratic equations. The ability to manipulate and rearrange equations into this standard form is a fundamental skill in algebra and is essential for solving a wide range of mathematical problems.

Final General Quadratic Form

After performing all the necessary algebraic manipulations, we have successfully transformed the given rational equation into the general quadratic form. The final equation is: 7x² - 40x - 12 = 0. This equation is now in the standard quadratic form ax² + bx + c = 0, where a = 7, b = -40, and c = -12. This transformation is a critical step in solving the original rational equation. By expressing it in this form, we can now apply various methods to find the values of x that satisfy the equation. Methods such as factoring, completing the square, or using the quadratic formula are all applicable at this stage. The general quadratic form provides a clear and structured representation that makes these solution techniques easier to implement. The coefficients a, b, and c are now readily identifiable, which is essential for applying these methods correctly. For example, if we were to use the quadratic formula, which is x = (-b ± √(b² - 4ac)) / (2a), we can directly substitute the values of a, b, and c into the formula to find the solutions for x. This transformation not only simplifies the process of finding solutions but also provides a standardized format for analyzing and comparing different quadratic equations. The ability to convert equations into this general form is a fundamental skill in algebra and is crucial for solving a wide range of mathematical problems.

Conclusion

In conclusion, we have successfully transformed the given rational equation (5x - 2)/2 - (19x + 6)/(2x) = (3x - 2)/4 into the general quadratic form 7x² - 40x - 12 = 0. This process involved several key algebraic steps, each vital for achieving the final result. We began by eliminating the fractions by finding the least common denominator (LCD) and multiplying each term by it. This step cleared the fractions, making the equation easier to manipulate. Next, we expanded the terms by distributing the coefficients, which removed the parentheses and allowed us to combine like terms. Combining like terms was the third step, where we grouped terms with the same variable and exponent, simplifying the equation further. Finally, we rearranged the equation into the general quadratic form ax² + bx + c = 0 by moving all terms to one side, leaving zero on the other side. This final form is crucial because it allows us to apply standard methods for solving quadratic equations, such as factoring, completing the square, or using the quadratic formula. The coefficients a, b, and c are now clearly identifiable, making it straightforward to apply these solution techniques. Mastering these transformation techniques is essential for anyone studying algebra. It provides a systematic approach to solving rational equations and other complex algebraic problems. The ability to manipulate equations and express them in standard forms is a fundamental skill that underpins much of mathematical problem-solving. By understanding and practicing these steps, students can build a solid foundation in algebra and improve their ability to tackle more advanced mathematical challenges. This example demonstrates the importance of careful and methodical algebraic manipulation in achieving a desired form, which is a valuable skill in many areas of mathematics and beyond.