Transforming Rational Equations To Quadratic Form A Step-by-Step Guide

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In this comprehensive guide, we will delve into the process of transforming a rational equation into its general quadratic form. Specifically, we will focus on the equation (5x - 2)/2 - (19x + 6)/(2x) = (3x - 2)/4. This involves a series of algebraic manipulations, including finding a common denominator, simplifying the equation, and rearranging terms to fit the standard quadratic form of ax^2 + bx + c = 0. Understanding this process is crucial for solving a wide range of mathematical problems and gaining a deeper understanding of algebraic equations. By the end of this guide, you will be equipped with the knowledge and skills to confidently tackle similar problems. Let’s break down each step involved in this transformation, ensuring clarity and accuracy every step of the way. This journey through algebraic manipulation will not only help you solve this particular equation but also enhance your overall problem-solving abilities in mathematics.

1. Initial Equation and Strategy

To begin, let's restate the given equation: (5x - 2)/2 - (19x + 6)/(2x) = (3x - 2)/4. Our primary goal is to eliminate the fractions and consolidate the equation into the general quadratic form, which is represented as ax^2 + bx + c = 0. This form is essential because it allows us to easily apply various methods for solving quadratic equations, such as factoring, completing the square, or using the quadratic formula. The strategy we'll employ involves finding the least common denominator (LCD) of all the fractions in the equation. This will enable us to clear the fractions by multiplying each term by the LCD, thereby simplifying the equation. The LCD is the smallest multiple that all denominators can divide into evenly, which in this case is 4x. By multiplying each term by 4x, we will effectively eliminate the denominators and pave the way for further simplification. This initial step is crucial, as it sets the stage for the subsequent algebraic manipulations. The elimination of fractions is a common technique in solving rational equations, and mastering it is a key skill in algebra.

2. Finding the Least Common Denominator (LCD)

Identifying the least common denominator (LCD) is a pivotal step in simplifying rational equations. In our equation, (5x - 2)/2 - (19x + 6)/(2x) = (3x - 2)/4, the denominators are 2, 2x, and 4. To find the LCD, we need to identify the smallest expression that is divisible by all these denominators. We begin by examining the numerical coefficients: 2 and 4. The smallest number divisible by both is 4. Next, we consider the variable component: x. Since 2x includes x, and 4 does not, we must include x in our LCD. Thus, combining the numerical and variable components, we arrive at the LCD of 4x. This means that 4x is the smallest expression that can be divided evenly by 2, 2x, and 4. Understanding how to find the LCD is crucial not only for this specific problem but also for handling a wide range of rational equations. The LCD allows us to eliminate fractions, which often simplifies the equation significantly and makes it easier to solve. By correctly identifying the LCD, we set the stage for the next step: multiplying each term in the equation by 4x.

3. Multiplying the Equation by the LCD

With the LCD identified as 4x, we now proceed to multiply each term in the equation (5x - 2)/2 - (19x + 6)/(2x) = (3x - 2)/4 by 4x. This step is crucial as it clears the fractions, transforming the equation into a more manageable form. Let's break down the multiplication for each term:

  1. 4x * (5x - 2)/2: Here, 4x is multiplied by the first term. The 4 in 4x is divided by the 2 in the denominator, resulting in 2x. This 2x is then multiplied by the numerator (5x - 2), yielding 2x(5x - 2).
  2. 4x * -(19x + 6)/(2x): In this term, 4x is multiplied by the second term. The 2x in the denominator cancels out with the x in 4x, leaving us with 2. This 2 is then multiplied by the negative of the numerator -(19x + 6), resulting in -2(19x + 6).
  3. 4x * (3x - 2)/4: For the right side of the equation, 4x is multiplied by (3x - 2)/4. The 4 in 4x cancels out with the 4 in the denominator, leaving us with x. This x is then multiplied by the numerator (3x - 2), giving us x(3x - 2).

After performing these multiplications, the equation transforms from a rational equation with fractions to a polynomial equation without fractions. This is a significant simplification, making it easier to proceed with further algebraic manipulations. The resulting equation sets the stage for expanding the terms and rearranging them into the general quadratic form.

4. Expanding and Simplifying

Having multiplied the equation by the LCD 4x, we now have: 2x(5x - 2) - 2(19x + 6) = x(3x - 2). The next step involves expanding these terms by applying the distributive property. This means multiplying each term inside the parentheses by the term outside.

  1. Expanding 2x(5x - 2): We multiply 2x by both 5x and -2. This gives us (2x * 5x) + (2x * -2), which simplifies to 10x^2 - 4x.
  2. Expanding -2(19x + 6): Here, we multiply -2 by both 19x and 6. This results in (-2 * 19x) + (-2 * 6), which simplifies to -38x - 12.
  3. Expanding x(3x - 2): We multiply x by both 3x and -2. This gives us (x * 3x) + (x * -2), which simplifies to 3x^2 - 2x.

Now, substituting these expanded terms back into the equation, we get: 10x^2 - 4x - 38x - 12 = 3x^2 - 2x. This equation is now free of parentheses, but it still contains multiple terms. The next step is to combine like terms on each side of the equation to further simplify it. Combining like terms involves adding or subtracting terms that have the same variable and exponent. This process reduces the number of terms in the equation, making it easier to rearrange into the general quadratic form.

5. Combining Like Terms

Following the expansion of the equation, we have 10x^2 - 4x - 38x - 12 = 3x^2 - 2x. Our next task is to combine like terms on both sides of the equation. Like terms are those that have the same variable raised to the same power. In this equation, we have terms with x^2, terms with x, and constant terms.

On the left side of the equation, we have:

  • A single x^2 term: 10x^2
  • Two x terms: -4x and -38x
  • One constant term: -12

Combining the x terms, we add -4x and -38x to get -42x. Thus, the left side of the equation simplifies to 10x^2 - 42x - 12.

On the right side of the equation, we have:

  • A single x^2 term: 3x^2
  • One x term: -2x

There are no like terms to combine on the right side, so it remains as 3x^2 - 2x.

Now, our equation looks like this: 10x^2 - 42x - 12 = 3x^2 - 2x. This simplified form is much easier to work with. The next step involves rearranging the terms to bring all of them to one side of the equation, setting the other side to zero. This is necessary to put the equation into the general quadratic form ax^2 + bx + c = 0, which is the standard form for solving quadratic equations.

6. Rearranging into General Quadratic Form

After combining like terms, our equation stands as 10x^2 - 42x - 12 = 3x^2 - 2x. To transform this into the general quadratic form ax^2 + bx + c = 0, we need to move all terms to one side of the equation, leaving zero on the other side. A common practice is to move all terms to the side where the x^2 coefficient is positive, which in this case is the left side. This helps to avoid dealing with negative leading coefficients, although it's not strictly necessary.

To move the terms from the right side to the left, we perform the following operations:

  1. Subtract 3x^2 from both sides: This eliminates the 3x^2 term on the right side and changes the equation to 10x^2 - 3x^2 - 42x - 12 = -2x.
  2. Add 2x to both sides: This eliminates the -2x term on the right side and changes the equation to 10x^2 - 3x^2 - 42x + 2x - 12 = 0.

Now, we combine the like terms on the left side:

  • Combine the x^2 terms: 10x^2 - 3x^2 = 7x^2
  • Combine the x terms: -42x + 2x = -40x

The equation now simplifies to 7x^2 - 40x - 12 = 0. This is the general quadratic form, where a = 7, b = -40, and c = -12. This form is crucial for applying various methods to solve the quadratic equation, such as the quadratic formula, factoring, or completing the square. By rearranging the equation into this standard form, we have successfully set the stage for finding the solutions (roots) of the equation.

7. Final Quadratic Form and Coefficients

Having meticulously performed the algebraic manipulations, we have successfully transformed the given rational equation into its general quadratic form. The final equation is: 7x^2 - 40x - 12 = 0. This equation fits the standard quadratic form ax^2 + bx + c = 0, where:

  • a = 7: This is the coefficient of the x^2 term.
  • b = -40: This is the coefficient of the x term.
  • c = -12: This is the constant term.

Identifying these coefficients is essential because they are used in various methods for solving quadratic equations. For example, in the quadratic formula (x = (-b ± √(b^2 - 4ac)) / (2a)), the values of a, b, and c are directly substituted to find the solutions for x. Similarly, when completing the square or attempting to factor the quadratic, these coefficients play a crucial role in the process. Understanding the significance of the general quadratic form and the ability to correctly identify the coefficients are fundamental skills in algebra. They provide a solid foundation for solving a wide range of quadratic equations and related problems. This concludes the transformation process, highlighting the importance of each step in achieving the final quadratic form.

8. Conclusion

In summary, we have successfully transformed the rational equation (5x - 2)/2 - (19x + 6)/(2x) = (3x - 2)/4 into its general quadratic form: 7x^2 - 40x - 12 = 0. This process involved several key steps, each requiring careful attention to detail and accuracy. We began by identifying the least common denominator (LCD) as 4x, which allowed us to eliminate the fractions by multiplying each term in the equation by the LCD. This step is crucial for simplifying rational equations and making them easier to manipulate.

Next, we expanded the resulting terms using the distributive property and combined like terms to further simplify the equation. This involved multiplying terms inside parentheses by terms outside, and then adding or subtracting terms with the same variable and exponent. These algebraic manipulations are fundamental skills in mathematics, and mastering them is essential for solving more complex equations.

Finally, we rearranged the terms to bring all of them to one side of the equation, setting the other side to zero. This step is necessary to put the equation into the general quadratic form ax^2 + bx + c = 0, which is the standard form for solving quadratic equations. We identified the coefficients as a = 7, b = -40, and c = -12, which are crucial for applying various methods to solve the quadratic equation.

By following these steps, we have not only solved this particular problem but also demonstrated a general method for transforming rational equations into quadratic form. This skill is invaluable for tackling a wide range of mathematical problems and gaining a deeper understanding of algebraic equations. The ability to confidently manipulate and transform equations is a hallmark of mathematical proficiency, and this guide has provided a clear and comprehensive approach to achieving that goal.