Solve The Equation:${ \frac{3}{4-x} = 2(x+1) }$

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Introduction

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In this article, we will delve into the world of algebra and solve a complex equation step by step. The equation we will be solving is $\frac{3}{4-x} = 2(x+1)$. This equation may seem daunting at first, but with a clear understanding of the steps involved, we can break it down into manageable parts and arrive at a solution.

Understanding the Equation

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Before we begin solving the equation, let's take a closer look at its structure. The equation is a rational equation, which means it contains fractions with variables in the denominator. The equation is also a quadratic equation, as it can be rewritten in the form of a quadratic equation.

The Equation in Standard Form

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To begin solving the equation, we need to rewrite it in standard form. The standard form of a quadratic equation is $ax^2 + bx + c = 0$. However, our equation is not in this form, so we need to manipulate it to get it into standard form.

Step 1: Multiply Both Sides by the Denominator

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The first step in solving the equation is to multiply both sides by the denominator, which is $4-x$. This will eliminate the fraction and allow us to work with a simpler equation.

$\frac{3}{4-x} = 2(x+1)$

Multiplying both sides by $4-x$, we get:

$3 = 2(x+1)(4-x)$

Expanding the Right-Hand Side

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Now that we have multiplied both sides by the denominator, we can expand the right-hand side of the equation.

$3 = 2(4-x)(x+1)$

Expanding the right-hand side, we get:

$3 = 2(4x + 4 - x^2 - x)$

Simplifying the right-hand side, we get:

$3 = 8x + 8 - 2x^2 - 2x$

Rearranging the Terms

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Now that we have expanded the right-hand side, we can rearrange the terms to get the equation into standard form.

$3 = -2x^2 + 6x + 8$

Step 2: Move All Terms to One Side

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The next step is to move all the terms to one side of the equation, so that we have a quadratic equation in standard form.

$-2x^2 + 6x + 8 - 3 = 0$

Simplifying the equation, we get:

$-2x^2 + 6x + 5 = 0$

Step 3: Solve the Quadratic Equation

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Now that we have the equation in standard form, we can solve it using the quadratic formula.

The quadratic formula is:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

In our equation, $a = -2$, $b = 6$, and $c = 5$. Plugging these values into the quadratic formula, we get:

$x = \frac{-6 \pm \sqrt{6^2 - 4(-2)(5)}}{2(-2)}$

Simplifying the equation, we get:

$x = \frac{-6 \pm \sqrt{36 + 40}}{-4}$

$x = \frac{-6 \pm \sqrt{76}}{-4}$

$x = \frac{-6 \pm 2\sqrt{19}}{-4}$

$x = \frac{3 \mp \sqrt{19}}{2}$

Conclusion

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In this article, we solved the equation $\frac{3}{4-x} = 2(x+1)$ step by step. We started by multiplying both sides by the denominator, then expanded the right-hand side, rearranged the terms, and finally solved the quadratic equation using the quadratic formula. The solutions to the equation are $x = \frac{3 \pm \sqrt{19}}{2}$.

Final Answer

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The final answer is $\boxed{\frac{3 \pm \sqrt{19}}{2}}$.

Discussion

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The equation we solved in this article is a complex rational equation. To solve it, we needed to multiply both sides by the denominator, expand the right-hand side, rearrange the terms, and finally solve the quadratic equation using the quadratic formula. The solutions to the equation are $x = \frac{3 \pm \sqrt{19}}{2}$.

Related Topics

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• Solving quadratic equations
• Rational equations
• Algebraic manipulations

References

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

Note: The references provided are for general information purposes only and are not directly related to the specific equation solved in this article.

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Introduction

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In our previous article, we solved the equation $\frac{3}{4-x} = 2(x+1)$ step by step. However, we understand that some readers may still have questions about the solution process. In this article, we will address some of the most frequently asked questions about solving the equation.

Q&A

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

A: The first step in solving the equation is to multiply both sides by the denominator, which is $4-x$. This will eliminate the fraction and allow us to work with a simpler equation.

Q: Why do we need to expand the right-hand side of the equation?

A: We need to expand the right-hand side of the equation to simplify it and make it easier to work with. Expanding the right-hand side allows us to combine like terms and rearrange the equation into standard form.

Q: What is the quadratic formula, and how do we use it to solve the equation?

A: The quadratic formula is a mathematical formula that allows us to solve quadratic equations. The quadratic formula is:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

We use the quadratic formula to solve the equation by plugging in the values of $a$, $b$, and $c$ into the formula.

Q: What are the solutions to the equation?

A: The solutions to the equation are $x = \frac{3 \pm \sqrt{19}}{2}$.

Q: How do we know that the solutions are correct?

A: We can verify the solutions by plugging them back into the original equation and checking that they satisfy the equation.

Q: What are some common mistakes to avoid when solving the equation?

A: Some common mistakes to avoid when solving the equation include:

• Not multiplying both sides by the denominator
• Not expanding the right-hand side of the equation
• Not using the quadratic formula correctly
• Not checking the solutions by plugging them back into the original equation

Q: What are some real-world applications of solving the equation?

A: Solving the equation has many real-world applications, including:

• Modeling population growth
• Analyzing financial data
• Solving optimization problems

Conclusion

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In this article, we addressed some of the most frequently asked questions about solving the equation $\frac{3}{4-x} = 2(x+1)$. We hope that this Q&A guide has been helpful in clarifying any confusion and providing a better understanding of the solution process.

Final Answer

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The final answer is $\boxed{\frac{3 \pm \sqrt{19}}{2}}$.

Discussion

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The equation we solved in this article is a complex rational equation. To solve it, we needed to multiply both sides by the denominator, expand the right-hand side, rearrange the terms, and finally solve the quadratic equation using the quadratic formula. The solutions to the equation are $x = \frac{3 \pm \sqrt{19}}{2}$.

Related Topics

---

• Solving quadratic equations
• Rational equations
• Algebraic manipulations

References

---

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

Note: The references provided are for general information purposes only and are not directly related to the specific equation solved in this article.