Solve The Rational Equation. Express Numbers As Integers Or Simplified Fractions.${\frac{28}{x^2}=2-\frac{1}{x}}$The Solution Set Is { \square$}$.

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Introduction

Rational equations are a fundamental concept in algebra, and solving them requires a combination of algebraic techniques and a deep understanding of fractions. In this article, we will explore the process of solving rational equations, with a focus on the given equation 28x2=2−1x\frac{28}{x^2}=2-\frac{1}{x}. We will break down the solution into manageable steps, using a combination of algebraic manipulations and fraction simplifications.

Understanding Rational Equations

A rational equation is an equation that contains fractions, where the numerator and denominator are polynomials. Rational equations can be solved using a variety of techniques, including cross-multiplication, factoring, and the quadratic formula. In this article, we will focus on the cross-multiplication method, which is a powerful tool for solving rational equations.

Cross-Multiplication Method

The cross-multiplication method involves multiplying both sides of the equation by the least common multiple (LCM) of the denominators. This eliminates the fractions and allows us to solve the equation using standard algebraic techniques. In the case of the given equation, the LCM of the denominators is x2x^2.

Step 1: Multiply Both Sides by the LCM

To eliminate the fractions, we multiply both sides of the equation by the LCM, which is x2x^2. This gives us:

28=x2(2−1x)28 = x^2\left(2-\frac{1}{x}\right)

Step 2: Distribute the x2x^2 Term

Next, we distribute the x2x^2 term to the terms inside the parentheses:

28=2x2−x28 = 2x^2 - x

Step 3: Move All Terms to One Side

To solve for xx, we need to move all the terms to one side of the equation. We can do this by subtracting 2x22x^2 from both sides and adding xx to both sides:

0=2x2−x−280 = 2x^2 - x - 28

Step 4: Factor the Quadratic Expression

The resulting equation is a quadratic expression, which can be factored using the quadratic formula or by finding two numbers that multiply to −56-56 and add to −1-1. In this case, we can factor the quadratic expression as:

0=(x+4)(2x−7)0 = (x + 4)(2x - 7)

Step 5: Solve for xx

To solve for xx, we set each factor equal to zero and solve for xx:

x+4=0⇒x=−4x + 4 = 0 \Rightarrow x = -4

2x−7=0⇒x=722x - 7 = 0 \Rightarrow x = \frac{7}{2}

Conclusion

In this article, we have solved the rational equation 28x2=2−1x\frac{28}{x^2}=2-\frac{1}{x} using the cross-multiplication method. We have broken down the solution into manageable steps, using a combination of algebraic manipulations and fraction simplifications. The final solution is x=−4x = -4 or x=72x = \frac{7}{2}.

Final Answer

The solution set is {−4,72}\boxed{\left\{-4, \frac{7}{2}\right\}}.

Tips and Tricks

  • When solving rational equations, it's essential to eliminate the fractions by multiplying both sides by the LCM.
  • Use the cross-multiplication method to eliminate fractions and simplify the equation.
  • Factor the quadratic expression to solve for xx.
  • Check your solutions by plugging them back into the original equation.

Common Mistakes

  • Failing to eliminate fractions by multiplying both sides by the LCM.
  • Not distributing the term correctly.
  • Not factoring the quadratic expression correctly.
  • Not checking solutions by plugging them back into the original equation.

Real-World Applications

Rational equations have numerous real-world applications, including:

  • Physics: Rational equations are used to describe the motion of objects under the influence of forces.
  • 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 trends.

Conclusion

In conclusion, solving rational equations requires a combination of algebraic techniques and a deep understanding of fractions. By following the steps outlined in this article, you can solve rational equations with ease. Remember to eliminate fractions by multiplying both sides by the LCM, distribute the term correctly, factor the quadratic expression, and check your solutions by plugging them back into the original equation. With practice and patience, you will become proficient in solving rational equations and be able to apply them to real-world problems.

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Introduction

In our previous article, we explored the process of solving rational equations using the cross-multiplication method. In this article, we will answer some of the most frequently asked questions about solving rational equations. Whether you're a student struggling to understand the concept or a teacher looking for ways to explain it to your students, this Q&A guide is for you.

Q: What is a rational equation?

A rational equation is an equation that contains fractions, where the numerator and denominator are polynomials. Rational equations can be solved using a variety of techniques, including cross-multiplication, factoring, and the quadratic formula.

Q: How do I know which method to use to solve a rational equation?

The choice of method depends on the complexity of the equation and the skills of the solver. Cross-multiplication is a good starting point for most rational equations, but factoring and the quadratic formula may be necessary for more complex equations.

Q: What is the least common multiple (LCM) of the denominators?

The LCM of the denominators is the smallest multiple that all the denominators have in common. In the case of the given equation, the LCM of the denominators is x2x^2.

Q: How do I eliminate fractions in a rational equation?

To eliminate fractions, multiply both sides of the equation by the LCM of the denominators. This will eliminate the fractions and allow you to solve the equation using standard algebraic techniques.

Q: What is the cross-multiplication method?

The cross-multiplication method involves multiplying both sides of the equation by the LCM of the denominators. This eliminates the fractions and allows you to solve the equation using standard algebraic techniques.

Q: How do I distribute the term in a rational equation?

To distribute the term, multiply the term by each of the terms inside the parentheses. For example, if the equation is x2(2−1x)x^2(2-\frac{1}{x}), you would multiply x2x^2 by 22 and x2x^2 by −1x-\frac{1}{x}.

Q: What is the quadratic formula?

The quadratic formula is a formula that can be used to solve quadratic equations. It is given by x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, where aa, bb, and cc are the coefficients of the quadratic equation.

Q: How do I factor a quadratic expression?

To factor a quadratic expression, look for two numbers that multiply to the constant term and add to the coefficient of the middle term. For example, if the quadratic expression is x2+5x+6x^2 + 5x + 6, you would look for two numbers that multiply to 66 and add to 55. In this case, the numbers are 22 and 33, so the factored form of the quadratic expression is (x+2)(x+3)(x + 2)(x + 3).

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

Some common mistakes to avoid when solving rational equations include:

  • Failing to eliminate fractions by multiplying both sides by the LCM.
  • Not distributing the term correctly.
  • Not factoring the quadratic expression correctly.
  • Not checking solutions by plugging them back into the original equation.

Q: How do I check my solutions to a rational equation?

To check your solutions, plug them back into the original equation and simplify. If the equation is true, then the solution is correct. If the equation is false, then the solution is incorrect.

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

Rational equations have numerous real-world applications, including:

  • Physics: Rational equations are used to describe the motion of objects under the influence of forces.
  • 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 trends.

Conclusion

In conclusion, solving rational equations requires a combination of algebraic techniques and a deep understanding of fractions. By following the steps outlined in this article and avoiding common mistakes, you can solve rational equations with ease. Remember to eliminate fractions by multiplying both sides by the LCM, distribute the term correctly, factor the quadratic expression, and check your solutions by plugging them back into the original equation. With practice and patience, you will become proficient in solving rational equations and be able to apply them to real-world problems.

Final Tips

  • Practice, practice, practice: The more you practice solving rational equations, the more comfortable you will become with the process.
  • Use online resources: There are many online resources available to help you learn and practice solving rational equations, including video tutorials and practice problems.
  • Seek help when needed: Don't be afraid to ask for help if you're struggling to understand a concept or solve a problem.