To find the x-intercept(s) of the equation \( y^2 = x^3 - 4x \), we need to determine the values of \( x \) when \( y = 0 \).

Let's solve the equation step by step:

1. Substitute \( y = 0 \) into the equation:
   \[
   0^2 = x^3 - 4x
   \]

2. Simplify the equation:
   \[
   0 = x^3 - 4x
   \]

3. Factor the equation:
   \[
   0 = x(x^2 - 4)
   \]
   \[
   0 = x(x - 2)(x + 2)
   \]

4. Solve for \( x \):
   \[
   x = 0 \quad \text{or} \quad x = 2 \quad \text{or} \quad x = -2
   \]

Therefore, the x-intercepts of the equation \( y^2 = x^3 - 4x \) are \( (0, 0) \), \( (2, 0) \), and \( (-2, 0) \).

Would you like more details or have any questions?

Here are 8 related questions to further explore this topic:

1. How do you find the vertex of the curve defined by \( y^2 = x^3 - 4x \)?
2. What are the symmetry properties of the curve \( y^2 = x^3 - 4x \)?
3. Can you find the points of intersection between \( y^2 = x^3 - 4x \) and another curve, such as \( y = x \)?
4. What is the general approach to finding the tangents to the curve \( y^2 = x^3 - 4x \)?
5. How do you determine if the curve \( y^2 = x^3 - 4x \) has any asymptotes?
6. How do you solve \( y^2 = x^3 - 4x \) for \( y \) in terms of \( x \)?
7. How does the curve \( y^2 = x^3 - 4x \) compare to a standard parabola or cubic function?
8. What is the significance of the discriminant in determining the nature of the roots of the equation \( x^3 - 4x = 0 \)?

**Tip:** When solving polynomial equations, factoring is a powerful tool for finding roots. If factoring is difficult, consider using the quadratic formula or other algebraic techniques.

Learn how to find the x-intercepts of the equation y^2 = x^3 - 4x with a detailed step-by-step solution. This problem involves polynomial equations and graphing techniques. Suitable for students in Grades 10-12.

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