To solve the equation \( x + \frac{1}{5}y^2 = 0 \) and find the equation of the parabola, we need to express \( x \) in terms of \( y \), or vice versa.

Starting with the given equation:

\[
x + \frac{1}{5}y^2 = 0
\]

1. Isolate \( x \):

\[
x = -\frac{1}{5}y^2
\]

This equation represents a parabola. The general form of a parabola is either \( y^2 = 4px \) (if the parabola opens to the left or right) or \( x^2 = 4py \) (if it opens upwards or downwards).

In this case, the equation can be rewritten as:

\[
y^2 = -5x
\]

Here, the parabola opens to the left because the coefficient of \( x \) is negative. The parabola is symmetric with respect to the x-axis, and the vertex of the parabola is at the origin \((0, 0)\).

### Key Features of the Parabola:
- **Vertex**: \((0, 0)\)
- **Axis of Symmetry**: The x-axis (\(y = 0\))
- **Direction of Opening**: Leftward, since the coefficient of \(x\) is negative.

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you determine the direction in which a parabola opens?
2. What is the significance of the vertex in a parabola?
3. How would the equation change if the parabola opened to the right?
4. Can you find the focus and directrix of the parabola from the given equation?
5. How does changing the coefficient of \(y^2\) affect the shape of the parabola?

**Tip**: For parabolas, the sign of the coefficient in front of \(x\) or \(y\) determines the direction in which the parabola opens.

Learn how to solve the parabola equation x + 1/5y² = 0 with a step-by-step solution. Understand the concept of parabolas and their equations, including vertex, axis of symmetry, and direction of opening. Suitable for high school students studying algebra.

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