https://file.aiquickdraw.com/mathbot/file/1725583781039m4b3ulwy.jpg

The image you uploaded shows the solution to a cubic equation. Let’s break down the steps presented in the image:

1. The given equation is:
   \[
   4x^3 - 4x = 0
   \]

2. **Factoring**: The equation is factored as:
   \[
   4x(x^2 - 1) = 0
   \]
   Here, \(4x\) is factored out from both terms of the cubic equation.

3. **Further Factorization**: The quadratic term \(x^2 - 1\) is recognized as a difference of squares:
   \[
   4x(x+1)(x-1) = 0
   \]
   This splits \(x^2 - 1\) into \((x+1)(x-1)\).

4. **Finding the Solutions**: Setting each factor equal to zero, we get the following solutions:
   \[
   x = 0, \quad x = -1, \quad x = 1
   \]

The three solutions to the equation are therefore \(x = 0\), \(x = -1\), and \(x = 1\).

### Do you have any specific questions about the solution or the method used?

Here are 5 related questions that could expand your understanding:

1. What is the difference of squares, and why is it important in factoring?
2. How can you identify whether a polynomial can be factored further?
3. Could the equation \(4x^3 - 4x = 0\) be solved using other methods, such as synthetic division?
4. How would the solution change if there were additional constants in the original equation?
5. What are the general strategies for solving cubic equations?

**Tip**: Always look for common factors first when solving polynomial equations; this simplifies the problem significantly!

Explore how to solve the cubic equation 4x^3 - 4x = 0 by factoring. This solution covers key algebraic concepts such as the difference of squares and the zero product property. Suitable for students in Grades 9-12.

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