Solve For $x$.$2 \sqrt[3]{x-3} = 4$
Introduction
In this article, we will delve into solving for x in the given equation 2 ∛(x-3) = 4. This equation involves a cube root, which can be solved using algebraic manipulations. We will break down the solution step by step, providing a clear understanding of the process.
Understanding the Equation
The given equation is 2 ∛(x-3) = 4. To solve for x, we need to isolate the variable x. The first step is to get rid of the coefficient 2 by dividing both sides of the equation by 2.
Step 1: Divide Both Sides by 2
∛(x-3) = 4/2 ∛(x-3) = 2
Step 2: Eliminate the Cube Root
To eliminate the cube root, we need to raise both sides of the equation to the power of 3.
(x-3) = 2^3 (x-3) = 8
Step 3: Solve for x
Now that we have eliminated the cube root, we can solve for x by adding 3 to both sides of the equation.
x - 3 + 3 = 8 + 3 x = 11
Conclusion
In this article, we have solved for x in the equation 2 ∛(x-3) = 4. By following the steps outlined above, we have isolated the variable x and found its value to be 11. This solution demonstrates the importance of algebraic manipulations in solving equations involving cube roots.
Tips and Tricks
- When solving equations involving cube roots, it is essential to eliminate the cube root by raising both sides of the equation to the power of 3.
- Always check your work by plugging the solution back into the original equation to ensure that it is true.
Real-World Applications
Solving equations involving cube roots has numerous real-world applications in fields such as engineering, physics, and computer science. For example, in engineering, cube roots are used to calculate the volume of irregularly shaped objects, while in physics, cube roots are used to calculate the energy of particles.
Common Mistakes
- Failing to eliminate the cube root by raising both sides of the equation to the power of 3.
- Not checking the solution by plugging it back into the original equation.
Practice Problems
- Solve for x in the equation ∛(x-2) = 3.
- Solve for x in the equation 2 ∛(x-1) = 5.
Conclusion
Introduction
In our previous article, we solved for x in the equation 2 ∛(x-3) = 4. In this article, we will address some common questions and concerns that readers may have regarding the solution.
Q: What is the cube root?
A: The cube root of a number is a value that, when multiplied by itself twice, gives the original number. For example, the cube root of 27 is 3, because 3 × 3 × 3 = 27.
Q: How do I eliminate the cube root?
A: To eliminate the cube root, you need to raise both sides of the equation to the power of 3. This will cancel out the cube root and allow you to solve for x.
Q: What if I have a negative number inside the cube root?
A: If you have a negative number inside the cube root, you need to be careful when raising both sides of the equation to the power of 3. This is because the cube of a negative number is negative. For example, if you have ∛(x-3) = 2, and x-3 is negative, then you need to be careful when raising both sides of the equation to the power of 3.
Q: Can I use a calculator to solve for x?
A: Yes, you can use a calculator to solve for x. However, it's always a good idea to check your work by plugging the solution back into the original equation to ensure that it is true.
Q: What if I have a fraction inside the cube root?
A: If you have a fraction inside the cube root, you need to be careful when raising both sides of the equation to the power of 3. This is because the cube of a fraction is not necessarily a fraction. For example, if you have ∛(x/3) = 2, then you need to be careful when raising both sides of the equation to the power of 3.
Q: Can I use the cube root to solve for x in a quadratic equation?
A: Yes, you can use the cube root to solve for x in a quadratic equation. However, you need to be careful when raising both sides of the equation to the power of 3, as this can lead to complex numbers.
Q: What if I have a complex number inside the cube root?
A: If you have a complex number inside the cube root, you need to be careful when raising both sides of the equation to the power of 3. This is because the cube of a complex number is not necessarily a complex number. For example, if you have ∛(x-3) = 2 + 3i, then you need to be careful when raising both sides of the equation to the power of 3.
Conclusion
Solving for x in the equation 2 ∛(x-3) = 4 requires careful algebraic manipulations. By following the steps outlined above, we have addressed some common questions and concerns that readers may have regarding the solution. We hope that this article has been helpful in clarifying any doubts that readers may have had.
Practice Problems
- Solve for x in the equation ∛(x-2) = 3.
- Solve for x in the equation 2 ∛(x-1) = 5.
- Solve for x in the equation ∛(x/3) = 2.
- Solve for x in the equation ∛(x-3) = 2 + 3i.
Tips and Tricks
- Always check your work by plugging the solution back into the original equation to ensure that it is true.
- Be careful when raising both sides of the equation to the power of 3, as this can lead to complex numbers.
- Use a calculator to check your work, but always double-check your solution by plugging it back into the original equation.