Enter The Correct Answer In The Box.Consider This Rational Equation: 1 X + X X − 2 = 1 4 \frac{1}{x} + \frac{x}{x-2} = \frac{1}{4} X 1 ​ + X − 2 X ​ = 4 1 ​ Use The Least Common Denominator To Simplify The Rational Equation Into A Standard Form Quadratic Equation. Replace The Values Of

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Introduction

Rational equations are a fundamental concept in algebra, and solving them requires a deep understanding of the underlying mathematics. In this article, we will explore the process of simplifying a rational equation using the least common denominator (LCD) and then converting it into a standard form quadratic equation. We will use the given rational equation 1x+xx−2=14\frac{1}{x} + \frac{x}{x-2} = \frac{1}{4} as a case study to illustrate the steps involved.

Understanding Rational Equations

A rational equation is an equation that contains one or more rational expressions, which are fractions that contain variables in the numerator or denominator. Rational equations can be solved using various techniques, including factoring, cross-multiplication, and the use of the LCD. In this article, we will focus on using the LCD to simplify the rational equation and then convert it into a standard form quadratic equation.

The Least Common Denominator (LCD)

The LCD is the smallest multiple of all the denominators in a rational equation. It is used to simplify the equation by eliminating the fractions. To find the LCD, we need to identify the denominators in the equation and then find their least common multiple. In the given equation, the denominators are xx and x−2x-2. The LCD is the product of these two denominators, which is x(x−2)x(x-2).

Simplifying the Rational Equation

To simplify the rational equation, we need to multiply both sides of the equation by the LCD. This will eliminate the fractions and allow us to work with a simpler equation. In this case, we multiply both sides of the equation by x(x−2)x(x-2).

1x+xx−2=14\frac{1}{x} + \frac{x}{x-2} = \frac{1}{4}

1x⋅x(x−2)+xx−2⋅x(x−2)=14⋅x(x−2)\frac{1}{x} \cdot x(x-2) + \frac{x}{x-2} \cdot x(x-2) = \frac{1}{4} \cdot x(x-2)

x(x−2)+x2=x(x−2)4x(x-2) + x^2 = \frac{x(x-2)}{4}

Converting to Standard Form Quadratic Equation

Now that we have simplified the rational equation, we can convert it into a standard form quadratic equation. To do this, we need to multiply both sides of the equation by 4 to eliminate the fraction.

4x(x−2)+4x2=x(x−2)4x(x-2) + 4x^2 = x(x-2)

4x2−8x+4x2=x2−2x4x^2 - 8x + 4x^2 = x^2 - 2x

8x2−8x=x2−2x8x^2 - 8x = x^2 - 2x

Rearranging the Terms

To put the equation into standard form, we need to rearrange the terms so that all the terms are on one side of the equation.

8x2−8x−x2+2x=08x^2 - 8x - x^2 + 2x = 0

7x2−6x=07x^2 - 6x = 0

Factoring the Quadratic Equation

Now that we have the equation in standard form, we can factor it to find the solutions. To do this, we need to find two numbers whose product is 77 and whose sum is −6-6. These numbers are −7-7 and −1-1, so we can factor the equation as follows:

7x2−6x=7x(x−67)7x^2 - 6x = 7x(x - \frac{6}{7})

Solving for x

To find the solutions, we need to set each factor equal to zero and solve for xx. In this case, we have two factors:

7x=07x = 0

x−67=0x - \frac{6}{7} = 0

Solving for xx in each equation, we get:

x=0x = 0

x=67x = \frac{6}{7}

Conclusion

In this article, we have explored the process of simplifying a rational equation using the least common denominator (LCD) and then converting it into a standard form quadratic equation. We used the given rational equation 1x+xx−2=14\frac{1}{x} + \frac{x}{x-2} = \frac{1}{4} as a case study to illustrate the steps involved. By following these steps, we were able to simplify the rational equation and convert it into a standard form quadratic equation, which we then factored to find the solutions. This process demonstrates the importance of using the LCD to simplify rational equations and the value of factoring to find the solutions.

Discussion Questions

  1. What is the least common denominator (LCD) of the rational equation 1x+xx−2=14\frac{1}{x} + \frac{x}{x-2} = \frac{1}{4}?
  2. How do you simplify a rational equation using the LCD?
  3. What is the standard form quadratic equation that results from simplifying the rational equation 1x+xx−2=14\frac{1}{x} + \frac{x}{x-2} = \frac{1}{4}?
  4. How do you factor a quadratic equation to find the solutions?
  5. What are the solutions to the rational equation 1x+xx−2=14\frac{1}{x} + \frac{x}{x-2} = \frac{1}{4}?

Additional Resources

For more information on solving rational equations and quadratic equations, please see the following resources:

  • Khan Academy: Solving Rational Equations
  • Mathway: Solving Quadratic Equations
  • Wolfram Alpha: Solving Rational Equations and Quadratic Equations

References

  • "Algebra" by Michael Artin
  • "College Algebra" by James Stewart
  • "Rational Equations and Functions" by Math Open Reference

Introduction

Rational equations are a fundamental concept in algebra, and solving them requires a deep understanding of the underlying mathematics. In this article, we will address some of the most frequently asked questions about rational equations, including how to simplify them, how to convert them into standard form quadratic equations, and how to find the solutions.

Q: What is a rational equation?

A: A rational equation is an equation that contains one or more rational expressions, which are fractions that contain variables in the numerator or denominator.

Q: How do I simplify a rational equation?

A: To simplify a rational equation, you need to find the least common denominator (LCD) of the fractions and multiply both sides of the equation by the LCD. This will eliminate the fractions and allow you to work with a simpler equation.

Q: What is the least common denominator (LCD)?

A: The LCD is the smallest multiple of all the denominators in a rational equation. It is used to simplify the equation by eliminating the fractions.

Q: How do I find the LCD?

A: To find the LCD, you need to identify the denominators in the equation and then find their least common multiple. You can use the following steps:

  1. Identify the denominators in the equation.
  2. Find the prime factorization of each denominator.
  3. Identify the common factors among the denominators.
  4. Multiply the common factors together to find the LCD.

Q: How do I convert a rational equation into a standard form quadratic equation?

A: To convert a rational equation into a standard form quadratic equation, you need to multiply both sides of the equation by the LCD and then rearrange the terms so that all the terms are on one side of the equation.

Q: How do I factor a quadratic equation?

A: To factor a quadratic equation, you need to find two numbers whose product is the constant term and whose sum is the coefficient of the linear term. These numbers are called the factors of the quadratic equation.

Q: What are the solutions to a rational equation?

A: The solutions to a rational equation are the values of the variable that make the equation true. To find the solutions, you need to set each factor equal to zero and solve for the variable.

Q: How do I check my solutions?

A: To check your solutions, you need to plug each solution back into the original equation and make sure that it is true. If the solution is true, then it is a valid solution to the equation.

Q: What are some common mistakes to avoid when solving rational equations?

A: Some common mistakes to avoid when solving rational equations include:

  • Not finding the LCD before simplifying the equation.
  • Not multiplying both sides of the equation by the LCD.
  • Not rearranging the terms so that all the terms are on one side of the equation.
  • Not factoring the quadratic equation correctly.
  • Not checking the solutions.

Q: What are some real-world applications of rational equations?

A: Rational equations have many real-world applications, including:

  • Physics: Rational equations are used to describe the motion of objects and the forces that act upon them.
  • Engineering: Rational equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Rational equations are used to model economic systems and make predictions about future economic trends.

Q: How can I practice solving rational equations?

A: There are many resources available to help you practice solving rational equations, including:

  • Online practice problems and quizzes.
  • Textbooks and workbooks.
  • Online tutoring and homework help services.
  • Practice tests and exams.

Conclusion

Rational equations are a fundamental concept in algebra, and solving them requires a deep understanding of the underlying mathematics. By following the steps outlined in this article, you can simplify rational equations, convert them into standard form quadratic equations, and find the solutions. Remember to check your solutions and avoid common mistakes when solving rational equations.

Additional Resources

For more information on solving rational equations, please see the following resources:

  • Khan Academy: Solving Rational Equations
  • Mathway: Solving Quadratic Equations
  • Wolfram Alpha: Solving Rational Equations and Quadratic Equations

References

  • "Algebra" by Michael Artin
  • "College Algebra" by James Stewart
  • "Rational Equations and Functions" by Math Open Reference