Solve The Equation:$\frac{36}{4x^2 + 36x} = 1 - \frac{1}{x + 9}$

Feb 26, 2025 by ADMIN 65 views

**Solving the Equation: A Step-by-Step Guide**

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Introduction

In this article, we will delve into the world of algebra and solve a complex equation step by step. The given equation is $\frac{36}{4x^2 + 36x} = 1 - \frac{1}{x + 9}$ . Our goal is to find the value of $x$ that satisfies this equation. We will break down the solution into manageable steps, making it easy to follow and understand.

Step 1: Simplify the Equation

The first step in solving the equation is to simplify it by getting rid of the fractions. To do this, we will multiply both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the LCM is $(x + 9)(4x^2 + 36x)$ .

\frac{36}{4x^2 + 36x} = 1 - \frac{1}{x + 9}

Multiplying both sides by $(x + 9)(4x^2 + 36x)$ , we get:

36(x + 9) = (x + 9)(4x^2 + 36x) - (4x^2 + 36x)

Step 2: Expand and Simplify

Next, we will expand and simplify the equation by multiplying the terms on the right-hand side.

36(x + 9) = 4x^3 + 36x^2 + 324x - 4x^2 - 36x

Combining like terms, we get:

36(x + 9) = 4x^3 + 32x^2 + 288x

Step 3: Distribute and Simplify

Now, we will distribute the $36$ on the left-hand side and simplify the equation.

36x + 324 = 4x^3 + 32x^2 + 288x

Subtracting $36x$ from both sides, we get:

324 = 4x^3 + 32x^2 + 252x

Step 4: Rearrange the Equation

The next step is to rearrange the equation to get all the terms on one side.

0 = 4x^3 + 32x^2 + 252x - 324

Step 5: Factor the Equation

Now, we will factor the equation to make it easier to solve.

0 = 4x(x^2 + 8x + 63) - 324

Factoring the quadratic expression inside the parentheses, we get:

0 = 4x(x + 3)(x + 21) - 324

Step 6: Solve for x

The final step is to solve for $x$ by setting each factor equal to zero.

4x(x + 3)(x + 21) - 324 = 0

Solving for $x$ , we get:

x = 0, x = -3, x = -21

Conclusion

In this article, we solved the equation $\frac{36}{4x^2 + 36x} = 1 - \frac{1}{x + 9}$ step by step. We simplified the equation, expanded and simplified, distributed and simplified, rearranged the equation, factored the equation, and finally solved for $x$ . The solutions to the equation are $x = 0, x = -3, x = -21$ .

Final Answer

The final answer is $\boxed{0, -3, -21}$ .

References

[1] Algebra, 2nd edition, by Michael Artin
[2] Calculus, 3rd edition, by Michael Spivak

Future Work

In the future, we can explore more complex equations and solve them using various techniques. We can also apply the solutions to real-world problems and see how they can be used to model and analyze complex systems.

Limitations

One limitation of this article is that it assumes a certain level of mathematical background and knowledge. Readers who are not familiar with algebra and calculus may find it difficult to follow. In the future, we can create more accessible and user-friendly content that caters to a wider range of readers.

Recommendations

Based on the solutions to the equation, we can make the following recommendations:

If $x = 0$ , then the equation is satisfied when $x$ is equal to zero.
If $x = -3$ , then the equation is satisfied when $x$ is equal to negative three.
If $x = -21$ , then the equation is satisfied when $x$ is equal to negative twenty-one.

These recommendations can be used to model and analyze complex systems in various fields, such as physics, engineering, and economics.

Future Research Directions

Some potential research directions include:

Solving equations with multiple variables
Applying the solutions to real-world problems
Developing new techniques for solving equations
Exploring the applications of algebra and calculus in various fields

By exploring these research directions, we can gain a deeper understanding of the solutions to complex equations and their applications in various fields.

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Introduction

In our previous article, we solved the equation $\frac{36}{4x^2 + 36x} = 1 - \frac{1}{x + 9}$ step by step. In this article, we will answer some frequently asked questions (FAQs) related to the solution of the equation.

Q: What is the final answer to the equation?

A: The final answer to the equation is $\boxed{0, -3, -21}$ .

Q: How do I simplify the equation?

A: To simplify the equation, you need to get rid of the fractions by multiplying both sides of the equation by the least common multiple (LCM) of the denominators.

Q: What is the LCM of the denominators?

A: The LCM of the denominators is $(x + 9)(4x^2 + 36x)$ .

Q: How do I expand and simplify the equation?

A: To expand and simplify the equation, you need to multiply the terms on the right-hand side and combine like terms.

Q: What is the final simplified equation?

A: The final simplified equation is $0 = 4x^3 + 32x^2 + 252x - 324$ .

Q: How do I factor the equation?

A: To factor the equation, you need to look for common factors and group them together.

Q: What is the factored equation?

A: The factored equation is $0 = 4x(x + 3)(x + 21) - 324$ .

Q: How do I solve for x?

A: To solve for x, you need to set each factor equal to zero and solve for x.

Q: What are the solutions to the equation?

A: The solutions to the equation are $x = 0, x = -3, x = -21$ .

Q: Can I apply the solutions to real-world problems?

A: Yes, you can apply the solutions to real-world problems. For example, you can use the solutions to model and analyze complex systems in physics, engineering, and economics.

Q: What are some potential research directions?

A: Some potential research directions include:

Solving equations with multiple variables
Applying the solutions to real-world problems
Developing new techniques for solving equations
Exploring the applications of algebra and calculus in various fields