Solve The Equation And Check For Extraneous Solutions:$6(4x - 1)^4 = 486$

Introduction

In this article, we will delve into the world of algebra and explore the process of solving a complex equation. The given equation is $6(4x - 1)^4 = 486$, and our goal is to find the value of $x$ that satisfies this equation. Additionally, we will check for any extraneous solutions that may arise during the process.

Understanding the Equation

The given equation is a quartic equation, which means it has a degree of 4. This type of equation can be challenging to solve, but with the right approach, we can break it down into manageable steps. The equation is $6(4x - 1)^4 = 486$, and our first step is to isolate the quartic term.

Isolating the Quartic Term

To isolate the quartic term, we need to divide both sides of the equation by 6. This gives us:

$$(4x - 1)^4 = 81$$

Now, we have a quartic equation in the form of $(4x - 1)^4 = 81$. Our next step is to take the fourth root of both sides of the equation.

Taking the Fourth Root

Taking the fourth root of both sides of the equation gives us:

$$4x - 1 = \sqrt[4]{81}$$

Since $\sqrt[4]{81} = 3$, we can simplify the equation to:

$$4x - 1 = 3$$

Solving for x

Now that we have a linear equation, we can solve for $x$. Adding 1 to both sides of the equation gives us:

$$4x = 4$$

Dividing both sides of the equation by 4 gives us:

$$x = 1$$

Checking for Extraneous Solutions

Before we conclude that $x = 1$ is the solution to the equation, we need to check for any extraneous solutions. An extraneous solution is a value of $x$ that satisfies the equation, but is not a valid solution.

To check for extraneous solutions, we need to plug the value of $x$ back into the original equation and verify that it is true. Plugging $x = 1$ into the original equation gives us:

$$6(4(1) - 1)^4 = 6(3)^4 = 6(81) = 486$$

Since the equation is true, we can conclude that $x = 1$ is the solution to the equation.

Conclusion

In this article, we solved the equation $6(4x - 1)^4 = 486$ and checked for extraneous solutions. We broke down the equation into manageable steps, isolated the quartic term, took the fourth root, and solved for $x$. We also checked for extraneous solutions by plugging the value of $x$ back into the original equation. The final solution to the equation is $x = 1$.

Tips and Tricks

When solving complex equations like this one, it's essential to break them down into manageable steps. Isolating the quartic term and taking the fourth root can help simplify the equation and make it easier to solve. Additionally, checking for extraneous solutions is crucial to ensure that the solution is valid.

Common Mistakes

When solving complex equations, it's easy to make mistakes. Some common mistakes include:

• Not isolating the quartic term
• Not taking the fourth root
• Not checking for extraneous solutions
• Not plugging the value of $x$ back into the original equation

By avoiding these common mistakes, you can ensure that your solution is accurate and valid.

Real-World Applications

Solving complex equations like this one has real-world applications in various fields, including:

• Physics: Solving complex equations is essential in physics, where equations are used to describe the behavior of particles and systems.
• Engineering: Solving complex equations is crucial in engineering, where equations are used to design and optimize systems.
• Computer Science: Solving complex equations is essential in computer science, where equations are used to develop algorithms and models.

By mastering the art of solving complex equations, you can apply your skills to real-world problems and make a meaningful impact in your field.

Conclusion

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Q&A: Frequently Asked Questions

Q: What is a quartic equation? A: A quartic equation is a polynomial equation of degree 4, which means it has a highest power of 4. Quartic equations can be challenging to solve, but with the right approach, we can break them down into manageable steps.

Q: How do I isolate the quartic term in an equation? A: To isolate the quartic term, we need to divide both sides of the equation by the coefficient of the quartic term. In the case of the equation $6(4x - 1)^4 = 486$, we divide both sides by 6 to get $(4x - 1)^4 = 81$.

Q: What is the fourth root of a number? A: The fourth root of a number is a value that, when raised to the power of 4, gives us the original number. For example, the fourth root of 81 is 3, because $3^4 = 81$.

Q: How do I check for extraneous solutions? A: To check for extraneous solutions, we need to plug the value of $x$ back into the original equation and verify that it is true. If the equation is not true, then the value of $x$ is an extraneous solution.

Q: What are some common mistakes to avoid when solving complex equations? A: Some common mistakes to avoid when solving complex equations include:

• Not isolating the quartic term
• Not taking the fourth root
• Not checking for extraneous solutions
• Not plugging the value of $x$ back into the original equation

Q: What are some real-world applications of solving complex equations? A: Solving complex equations has real-world applications in various fields, including:

• Physics: Solving complex equations is essential in physics, where equations are used to describe the behavior of particles and systems.
• Engineering: Solving complex equations is crucial in engineering, where equations are used to design and optimize systems.
• Computer Science: Solving complex equations is essential in computer science, where equations are used to develop algorithms and models.

Q: How can I practice solving complex equations? A: To practice solving complex equations, you can try solving problems on your own or using online resources such as Khan Academy or Wolfram Alpha. You can also try solving problems from textbooks or online resources.

Q: What are some tips for mastering the art of solving complex equations? A: Some tips for mastering the art of solving complex equations include:

• Breaking down the equation into manageable steps
• Isolating the quartic term
• Taking the fourth root
• Checking for extraneous solutions
• Practicing regularly

Q: Can I use a calculator to solve complex equations? A: Yes, you can use a calculator to solve complex equations. However, it's essential to understand the underlying math and be able to explain the steps involved in solving the equation.

Q: What are some common types of complex equations? A: Some common types of complex equations include:

• Quartic equations
• Cubic equations
• Quadratic equations
• Linear equations

Q: How can I apply my skills in solving complex equations to real-world problems? A: You can apply your skills in solving complex equations to real-world problems by:

• Using equations to model real-world systems
• Developing algorithms and models to solve complex problems
• Designing and optimizing systems using equations
• Analyzing data and making predictions using equations

Conclusion

In conclusion, solving complex equations requires a step-by-step approach and a deep understanding of the underlying math. By mastering the art of solving complex equations, you can apply your skills to real-world problems and make a meaningful impact in your field. Remember to practice regularly, break down the equation into manageable steps, and check for extraneous solutions.