Which Of The Following Occurs Within The Solution Process For $\sqrt[3]{5x-2} - \sqrt[3]{4x} = 0$?A. Squaring Both Sides OnceB. Squaring Both Sides TwiceC. Cubing Both Sides OnceD. Cubing Both Sides Twice

Introduction

In this article, we will explore the solution process for the cubic equation $\sqrt[3]{5x-2} - \sqrt[3]{4x} = 0$. This equation involves the use of cube roots and requires a careful approach to solve. We will examine the different options for solving this equation and determine which one is the correct solution process.

Understanding the Equation

The given equation is $\sqrt[3]{5x-2} - \sqrt[3]{4x} = 0$. To solve this equation, we need to isolate the variable x. The first step is to get rid of the cube roots. We can do this by cubing both sides of the equation.

Cubing Both Sides

Cubing both sides of the equation is a common technique used to eliminate cube roots. When we cube both sides, we get:

$$\left(\sqrt[3]{5x-2} - \sqrt[3]{4x}\right)^3 = 0^3$$

Expanding the left-hand side of the equation, we get:

$$(5x-2)^3 - 3(5x-2)^2\sqrt[3]{4x} + 3(5x-2)\left(\sqrt[3]{4x}\right)^2 - \left(\sqrt[3]{4x}\right)^3 = 0$$

Simplifying the equation, we get:

$$(5x-2)^3 - 3(5x-2)^2\sqrt[3]{4x} + 3(5x-2)\left(\sqrt[3]{4x}\right)^2 - \left(\sqrt[3]{4x}\right)^3 = 0$$

Squaring Both Sides

Another option for solving this equation is to square both sides. However, squaring both sides once is not enough to eliminate the cube roots. We need to square both sides twice to get rid of the cube roots.

Squaring Both Sides Twice

Squaring both sides twice is a more complex process. When we square both sides twice, we get:

$$\left(\sqrt[3]{5x-2} - \sqrt[3]{4x}\right)^2 = 0^2$$

Expanding the left-hand side of the equation, we get:

$$(5x-2)^2 - 2(5x-2)\sqrt[3]{4x} + \left(\sqrt[3]{4x}\right)^2 = 0$$

Squaring both sides again, we get:

$$\left((5x-2)^2 - 2(5x-2)\sqrt[3]{4x} + \left(\sqrt[3]{4x}\right)^2\right)^2 = 0^2$$

Simplifying the equation, we get:

$$(5x-2)^4 - 4(5x-2)^3\sqrt[3]{4x} + 6(5x-2)^2\left(\sqrt[3]{4x}\right)^2 - 4(5x-2)\left(\sqrt[3]{4x}\right)^3 + \left(\sqrt[3]{4x}\right)^4 = 0$$

Conclusion

In conclusion, the correct solution process for the cubic equation $\sqrt[3]{5x-2} - \sqrt[3]{4x} = 0$ is to cube both sides once. This is because cubing both sides once eliminates the cube roots and allows us to solve for the variable x. Squaring both sides once is not enough to eliminate the cube roots, and squaring both sides twice is a more complex process that is not necessary in this case.

Answer

The correct answer is:

• C. Cubing both sides once

Q: What is a cubic equation?

A: A cubic equation is a polynomial equation of degree three, which means that the highest power of the variable (in this case, x) is three. Cubic equations can be written in the form ax^3 + bx^2 + cx + d = 0, where a, b, c, and d are constants.

Q: How do I solve a cubic equation?

A: There are several methods for solving cubic equations, including factoring, the rational root theorem, and the cubic formula. However, the cubic formula is the most general method and is used when the equation cannot be factored or solved using the rational root theorem.

Q: What is the cubic formula?

A: The cubic formula is a general method for solving cubic equations. It is given by:

x = (-b + sqrt(b^2 - 3ac)) / (3a)

where a, b, and c are the coefficients of the cubic equation.

Q: How do I use the cubic formula?

A: To use the cubic formula, you need to identify the coefficients a, b, and c in the cubic equation. Then, you plug these values into the formula and simplify to find the value of x.

Q: What are some common mistakes to avoid when solving cubic equations?

A: Some common mistakes to avoid when solving cubic equations include:

• Not checking for extraneous solutions
• Not simplifying the equation before solving
• Not using the correct formula for the specific type of equation
• Not checking the solution for validity

Q: Can I use a calculator to solve cubic equations?

A: Yes, you can use a calculator to solve cubic equations. However, it's always a good idea to check the solution by hand to make sure it's correct.

Q: How do I check my solution for validity?

A: To check your solution for validity, you need to plug the value of x back into the original equation and simplify. If the equation is true, then the solution is valid. If the equation is false, then the solution is not valid.

Q: What are some real-world applications of cubic equations?

A: Cubic equations have many real-world applications, including:

• Physics: Cubic equations are used to model the motion of objects under the influence of gravity.
• Engineering: Cubic equations are used to design and optimize systems, such as bridges and buildings.
• Computer Science: Cubic equations are used in computer graphics and game development.

Q: Can I use cubic equations to solve other types of equations?

A: Yes, you can use cubic equations to solve other types of equations, including quadratic equations and polynomial equations of higher degree. However, the method of solution will depend on the specific type of equation.

Q: How do I know which method to use to solve a cubic equation?

A: The method you use to solve a cubic equation will depend on the specific equation and the values of the coefficients. In general, you should try to factor the equation first, and then use the rational root theorem or the cubic formula if necessary.

Q: What are some common types of cubic equations?

A: Some common types of cubic equations include:

• Monic cubic equations: These are cubic equations of the form x^3 + bx^2 + cx + d = 0.
• Non-monic cubic equations: These are cubic equations of the form ax^3 + bx^2 + cx + d = 0, where a is not equal to 1.
• Irreducible cubic equations: These are cubic equations that cannot be factored into the product of two binomials.

Q: Can I use cubic equations to solve systems of equations?

A: Yes, you can use cubic equations to solve systems of equations. However, the method of solution will depend on the specific system of equations and the values of the coefficients.

Q: How do I use cubic equations to solve systems of equations?

A: To use cubic equations to solve systems of equations, you need to first write the system of equations as a single cubic equation. Then, you can use the methods described above to solve the cubic equation and find the values of the variables.