Solve The Equation:${ \frac{2x}{2+x} = \frac{1}{x-4} }$

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Introduction

In this article, we will delve into the world of algebra and solve a complex equation involving fractions. The equation we will be solving is 2x2+x=1xโˆ’4\frac{2x}{2+x} = \frac{1}{x-4}. This equation may seem daunting at first, but with a systematic approach and a few clever manipulations, we can simplify it and find the solution.

Understanding the Equation

Before we begin solving the equation, let's take a closer look at its structure. We have two fractions on either side of the equation, each with a different denominator. The left-hand side has a denominator of 2+x2+x, while the right-hand side has a denominator of xโˆ’4x-4. Our goal is to eliminate these denominators and solve for the variable xx.

Step 1: Cross-Multiplication

One way to eliminate the denominators is to cross-multiply the fractions. This means multiplying the numerator of the left-hand fraction by the denominator of the right-hand fraction, and vice versa. This gives us:

2x(xโˆ’4)=1(2+x)2x(x-4) = 1(2+x)

Step 2: Expanding and Simplifying

Now that we have cross-multiplied, we can expand and simplify the equation. Expanding the left-hand side gives us:

2x2โˆ’8x=2+x2x^2 - 8x = 2 + x

Step 3: Rearranging the Equation

To make it easier to solve for xx, let's rearrange the equation by moving all the terms to one side. This gives us:

2x2โˆ’9xโˆ’2=02x^2 - 9x - 2 = 0

Step 4: Factoring the Quadratic

Now that we have a quadratic equation, let's try to factor it. Factoring a quadratic equation involves finding two numbers whose product is the constant term and whose sum is the coefficient of the linear term. In this case, the constant term is โˆ’2-2 and the coefficient of the linear term is โˆ’9-9. After some trial and error, we find that the factored form of the quadratic is:

(2x+1)(xโˆ’2)=0(2x + 1)(x - 2) = 0

Step 5: Solving for xx

Now that we have factored the quadratic, we can solve for xx by setting each factor equal to zero. This gives us two possible solutions:

2x+1=0โ‡’x=โˆ’122x + 1 = 0 \Rightarrow x = -\frac{1}{2}

xโˆ’2=0โ‡’x=2x - 2 = 0 \Rightarrow x = 2

Conclusion

And there you have it! We have solved the equation 2x2+x=1xโˆ’4\frac{2x}{2+x} = \frac{1}{x-4} using a combination of cross-multiplication, expansion, simplification, rearrangement, factoring, and solving for xx. The two solutions to the equation are x=โˆ’12x = -\frac{1}{2} and x=2x = 2. We hope this article has provided a clear and concise guide to solving this complex equation.

Final Thoughts

Solving equations like this one requires a combination of mathematical techniques and problem-solving skills. By breaking down the equation into smaller steps and using a systematic approach, we can simplify it and find the solution. We hope this article has inspired you to tackle more complex equations and explore the world of algebra.

Additional Resources

If you're interested in learning more about algebra and solving equations, here are some additional resources you may find helpful:

  • Khan Academy: Algebra
  • MIT OpenCourseWare: Algebra
  • Wolfram Alpha: Algebra Solver

Glossary of Terms

Here are some key terms and concepts used in this article:

  • Cross-multiplication: A technique used to eliminate denominators in fractions by multiplying the numerator of one fraction by the denominator of the other fraction.
  • Quadratic equation: An equation of the form ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants.
  • Factoring: A technique used to simplify quadratic equations by expressing them as a product of two binomials.
  • Solving for xx: The process of finding the value of the variable xx that satisfies an equation.

References

  • [1] "Algebra" by Michael Artin
  • [2] "Calculus" by Michael Spivak
  • [3] "Linear Algebra" by Jim Hefferon

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Introduction

In our previous article, we solved the equation 2x2+x=1xโˆ’4\frac{2x}{2+x} = \frac{1}{x-4} using a combination of mathematical techniques and problem-solving skills. In this article, we will provide a Q&A guide to help you better understand the solution and answer any questions you may have.

Q: What is the first step in solving the equation?

A: The first step in solving the equation is to cross-multiply the fractions. This involves multiplying the numerator of the left-hand fraction by the denominator of the right-hand fraction, and vice versa.

Q: Why do we need to cross-multiply?

A: We need to cross-multiply to eliminate the denominators in the fractions. This makes it easier to simplify the equation and solve for the variable xx.

Q: What is the next step after cross-multiplication?

A: After cross-multiplication, we need to expand and simplify the equation. This involves multiplying out the terms and combining like terms.

Q: How do we expand and simplify the equation?

A: To expand and simplify the equation, we need to multiply out the terms and combine like terms. For example, if we have the equation 2x2โˆ’8x=2+x2x^2 - 8x = 2 + x, we can expand and simplify it by combining the like terms.

Q: What is the next step after expanding and simplifying?

A: After expanding and simplifying, we need to rearrange the equation by moving all the terms to one side. This makes it easier to solve for the variable xx.

Q: How do we rearrange the equation?

A: To rearrange the equation, we need to move all the terms to one side by adding or subtracting the same value from both sides. For example, if we have the equation 2x2โˆ’9xโˆ’2=02x^2 - 9x - 2 = 0, we can rearrange it by moving all the terms to the left-hand side.

Q: What is the next step after rearranging the equation?

A: After rearranging the equation, we need to factor the quadratic expression. This involves expressing the quadratic expression as a product of two binomials.

Q: How do we factor the quadratic expression?

A: To factor the quadratic expression, we need to find two numbers whose product is the constant term and whose sum is the coefficient of the linear term. For example, if we have the quadratic expression 2x2โˆ’9xโˆ’22x^2 - 9x - 2, we can factor it by finding two numbers whose product is โˆ’2-2 and whose sum is โˆ’9-9.

Q: What is the final step in solving the equation?

A: The final step in solving the equation is to solve for the variable xx by setting each factor equal to zero.

Q: How do we solve for the variable xx?

A: To solve for the variable xx, we need to set each factor equal to zero and solve for xx. For example, if we have the equation (2x+1)(xโˆ’2)=0(2x + 1)(x - 2) = 0, we can solve for xx by setting each factor equal to zero.

Conclusion

We hope this Q&A guide has helped you better understand the solution to the equation 2x2+x=1xโˆ’4\frac{2x}{2+x} = \frac{1}{x-4}. If you have any further questions or need additional help, please don't hesitate to ask.

Additional Resources

If you're interested in learning more about algebra and solving equations, here are some additional resources you may find helpful:

  • Khan Academy: Algebra
  • MIT OpenCourseWare: Algebra
  • Wolfram Alpha: Algebra Solver

Glossary of Terms

Here are some key terms and concepts used in this article:

  • Cross-multiplication: A technique used to eliminate denominators in fractions by multiplying the numerator of one fraction by the denominator of the other fraction.
  • Quadratic equation: An equation of the form ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants.
  • Factoring: A technique used to simplify quadratic equations by expressing them as a product of two binomials.
  • Solving for xx: The process of finding the value of the variable xx that satisfies an equation.

References

  • [1] "Algebra" by Michael Artin
  • [2] "Calculus" by Michael Spivak
  • [3] "Linear Algebra" by Jim Hefferon