Solve The Rational Equation:${ \frac{2x^2+1}{x-1} + X = 3 + \frac{3}{x-1} }$A. There Is No Solution. B. { X = \frac{1}{3}, X = 1 $}$ C. { X = \frac{1}{3} $}$ D. { X = 1 $}$

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Rational equations are a fundamental concept in algebra, and solving them requires a combination of algebraic techniques and a deep understanding of the underlying mathematics. In this article, we will explore the process of solving rational equations, using the given equation as a case study.

What are Rational Equations?

A rational equation is an equation that contains one or more rational expressions, which are expressions that can be written in the form of a fraction. Rational expressions can be added, subtracted, multiplied, and divided, just like regular expressions. However, when working with rational expressions, we must be mindful of the restrictions on the domain, which are the values of the variable that make the expression undefined.

The Given Equation

The given equation is:

2x2+1x−1+x=3+3x−1\frac{2x^2+1}{x-1} + x = 3 + \frac{3}{x-1}

This equation contains two rational expressions: 2x2+1x−1\frac{2x^2+1}{x-1} and 3x−1\frac{3}{x-1}. The equation also contains a linear term, xx, and a constant term, 33.

Step 1: Simplify the Equation

To simplify the equation, we can start by combining the rational expressions on the left-hand side. We can do this by finding a common denominator, which is x−1x-1.

2x2+1x−1+x=2x2+1x−1+x(x−1)x−1\frac{2x^2+1}{x-1} + x = \frac{2x^2+1}{x-1} + \frac{x(x-1)}{x-1}

Simplifying the right-hand side, we get:

2x2+1x−1+x=2x2+1+x2−xx−1\frac{2x^2+1}{x-1} + x = \frac{2x^2+1+x^2-x}{x-1}

Step 2: Combine Like Terms

We can now combine like terms on the numerator:

2x2+1+x2−xx−1=3x2−x+1x−1\frac{2x^2+1+x^2-x}{x-1} = \frac{3x^2-x+1}{x-1}

Step 3: Equate the Numerators

Since the denominators are the same, we can equate the numerators:

3x2−x+1=3(x−1)+33x^2-x+1 = 3(x-1) + 3

Step 4: Expand and Simplify

Expanding the right-hand side, we get:

3x2−x+1=3x−3+33x^2-x+1 = 3x-3+3

Simplifying further, we get:

3x2−x+1=3x3x^2-x+1 = 3x

Step 5: Move All Terms to One Side

To solve for xx, we can move all terms to one side of the equation:

3x2−x+1−3x=03x^2-x+1-3x = 0

Simplifying further, we get:

3x2−4x+1=03x^2-4x+1 = 0

Step 6: Solve the Quadratic Equation

This is a quadratic equation in the form of ax2+bx+c=0ax^2+bx+c=0. We can solve it using the quadratic formula:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2-4ac}}{2a}

In this case, a=3a=3, b=−4b=-4, and c=1c=1. Plugging these values into the formula, we get:

x=−(−4)±(−4)2−4(3)(1)2(3)x = \frac{-(-4) \pm \sqrt{(-4)^2-4(3)(1)}}{2(3)}

Simplifying further, we get:

x=4±16−126x = \frac{4 \pm \sqrt{16-12}}{6}

x=4±46x = \frac{4 \pm \sqrt{4}}{6}

x=4±26x = \frac{4 \pm 2}{6}

Step 7: Simplify the Solutions

We now have two possible solutions:

x=4+26=66=1x = \frac{4+2}{6} = \frac{6}{6} = 1

x=4−26=26=13x = \frac{4-2}{6} = \frac{2}{6} = \frac{1}{3}

Conclusion

In this article, we have solved the rational equation 2x2+1x−1+x=3+3x−1\frac{2x^2+1}{x-1} + x = 3 + \frac{3}{x-1} using a step-by-step approach. We have simplified the equation, combined like terms, equated the numerators, expanded and simplified, moved all terms to one side, and solved the quadratic equation. The final solutions are x=1x = 1 and x=13x = \frac{1}{3}.

Final Answer

The final answer is:

  • x=13x = \frac{1}{3} and x=1x = 1

Note: The final answer is a combination of the two solutions obtained in the previous step.

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In the previous article, we solved the rational equation 2x2+1x−1+x=3+3x−1\frac{2x^2+1}{x-1} + x = 3 + \frac{3}{x-1} using a step-by-step approach. However, we may still have some questions about the process and the solutions obtained. In this article, we will address some of the most frequently asked questions about rational equation solutions.

Q: What is the difference between a rational equation and a rational expression?

A: A rational expression is an expression that can be written in the form of a fraction, such as 2x2+1x−1\frac{2x^2+1}{x-1}. A rational equation, on the other hand, is an equation that contains one or more rational expressions, such as 2x2+1x−1+x=3+3x−1\frac{2x^2+1}{x-1} + x = 3 + \frac{3}{x-1}.

Q: How do I know if a rational equation has a solution?

A: To determine if a rational equation has a solution, we need to check if the equation is true for all values of the variable. If the equation is true for all values of the variable, then it has a solution. However, if the equation is not true for all values of the variable, then it may not have a solution.

Q: What is the process for solving a rational equation?

A: The process for solving a rational equation involves the following steps:

  1. Simplify the equation by combining like terms and eliminating any common factors.
  2. Equate the numerators of the rational expressions.
  3. Expand and simplify the resulting equation.
  4. Move all terms to one side of the equation.
  5. Solve the resulting equation using algebraic techniques.

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

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

  • Not simplifying the equation before solving it.
  • Not equating the numerators of the rational expressions.
  • Not expanding and simplifying the resulting equation.
  • Not moving all terms to one side of the equation.
  • Not checking for extraneous solutions.

Q: How do I check for extraneous solutions?

A: To check for extraneous solutions, we need to plug the solution back into the original equation and check if it is true. If the solution is not true, then it is an extraneous solution and should be discarded.

Q: What is the difference between a rational equation and a quadratic equation?

A: A rational equation is an equation that contains one or more rational expressions, such as 2x2+1x−1+x=3+3x−1\frac{2x^2+1}{x-1} + x = 3 + \frac{3}{x-1}. A quadratic equation, on the other hand, is an equation that can be written in the form of ax2+bx+c=0ax^2+bx+c=0, such as 3x2−4x+1=03x^2-4x+1=0.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, we can use the quadratic formula:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2-4ac}}{2a}

We can also use factoring, completing the square, or the quadratic formula to solve a quadratic equation.

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 behavior of physical systems.
  • 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 economic behavior.
  • Computer Science: Rational equations are used to solve problems in computer science, such as graph theory and network analysis.

Q: How do I know if a rational equation has a real-world application?

A: To determine if a rational equation has a real-world application, we need to consider the context in which the equation is being used. If the equation is being used to model a real-world system or phenomenon, then it likely has a real-world application.

Q: What are some common types of rational equations?

A: Some common types of rational equations include:

  • Linear rational equations: These are rational equations that contain a linear term, such as 2x2+1x−1+x=3+3x−1\frac{2x^2+1}{x-1} + x = 3 + \frac{3}{x-1}.
  • Quadratic rational equations: These are rational equations that contain a quadratic term, such as 3x2−4x+1=03x^2-4x+1=0.
  • Polynomial rational equations: These are rational equations that contain a polynomial term, such as x3+2x2−3x+1x−1=2x2+3x−1\frac{x^3+2x^2-3x+1}{x-1} = 2x^2+3x-1.

Q: How do I choose the correct method for solving a rational equation?

A: To choose the correct method for solving a rational equation, we need to consider the type of equation and the level of difficulty. If the equation is a simple linear rational equation, we can use the method of equating the numerators. If the equation is a more complex rational equation, we may need to use a combination of methods, such as factoring and the quadratic formula.

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

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

  • Not simplifying the equation before solving it.
  • Not equating the numerators of the rational expressions.
  • Not expanding and simplifying the resulting equation.
  • Not moving all terms to one side of the equation.
  • Not checking for extraneous solutions.

Q: How do I check for extraneous solutions?

A: To check for extraneous solutions, we need to plug the solution back into the original equation and check if it is true. If the solution is not true, then it is an extraneous solution and should be discarded.

Q: What is the difference between a rational equation and a polynomial equation?

A: A rational equation is an equation that contains one or more rational expressions, such as 2x2+1x−1+x=3+3x−1\frac{2x^2+1}{x-1} + x = 3 + \frac{3}{x-1}. A polynomial equation, on the other hand, is an equation that contains a polynomial term, such as x3+2x2−3x+1=0x^3+2x^2-3x+1=0.

Q: How do I solve a polynomial equation?

A: To solve a polynomial equation, we can use a combination of methods, such as factoring, the quadratic formula, and synthetic division.

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 behavior of physical systems.
  • 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 economic behavior.
  • Computer Science: Rational equations are used to solve problems in computer science, such as graph theory and network analysis.

Q: How do I know if a rational equation has a real-world application?

A: To determine if a rational equation has a real-world application, we need to consider the context in which the equation is being used. If the equation is being used to model a real-world system or phenomenon, then it likely has a real-world application.

Q: What are some common types of rational equations?

A: Some common types of rational equations include:

  • Linear rational equations: These are rational equations that contain a linear term, such as 2x2+1x−1+x=3+3x−1\frac{2x^2+1}{x-1} + x = 3 + \frac{3}{x-1}.
  • Quadratic rational equations: These are rational equations that contain a quadratic term, such as 3x2−4x+1=03x^2-4x+1=0.
  • Polynomial rational equations: These are rational equations that contain a polynomial term, such as x3+2x2−3x+1x−1=2x2+3x−1\frac{x^3+2x^2-3x+1}{x-1} = 2x^2+3x-1.

Q: How do I choose the correct method for solving a rational equation?

A: To choose the correct method for solving a rational equation, we need to consider the type of equation and the level of difficulty. If the equation is a simple linear rational equation, we can use the method of equating the numerators. If the equation is a more complex rational equation, we may need to use a combination of methods, such as factoring and the quadratic formula.

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

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

  • Not simplifying the equation before solving it.
  • Not equating the numerators of the rational expressions.
  • Not expanding and simplifying the resulting equation.
  • Not moving all terms to one side of the equation.
  • Not checking for extraneous solutions.

Q: How do I check for extraneous solutions?

A: To check for extraneous solutions, we need to plug the solution back into the original equation and check if it is true. If the solution is not true, then it is an extraneous solution and should be discarded.

Q: What is the difference between a rational equation and a trigonometric equation?

A: A rational equation is an equation that contains one or more rational expressions, such as 2x2+1x−1+x=3+3x−1\frac{2x^2+1}{x-1} + x = 3 + \frac{3}{x-1}.