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To solve this problem, let’s analyze the given information step-by-step.

1. **Identify the Quadratic Function and Its Vertex**:
   The function provided is:
   \[
   y = -x^2 + 4x - 4
   \]
   This is a quadratic equation, and its graph will be a parabola that opens downward (since the coefficient of \( x^2 \) is negative).

2. **Condition of the Vertex on the x-axis**:
   It is stated that the vertex lies on the x-axis. This means the \( y \)-coordinate of the vertex must be zero.

3. **Find the Vertex of the Parabola**:
   For a quadratic function in the form \( y = ax^2 + bx + c \), the \( x \)-coordinate of the vertex is given by:
   \[
   x = -\frac{b}{2a}
   \]
   For \( y = -x^2 + 4x - 4 \), we have:
   - \( a = -1 \)
   - \( b = 4 \)

   Substituting these values:
   \[
   x = -\frac{4}{2 \times -1} = 2
   \]

   So, the \( x \)-coordinate of the vertex is \( x = 2 \).

4. **Determine the \( y \)-coordinate of the Vertex**:
   Substitute \( x = 2 \) back into the equation to find the \( y \)-coordinate:
   \[
   y = -(2)^2 + 4 \cdot 2 - 4 = -4 + 8 - 4 = 0
   \]

   Since the \( y \)-coordinate of the vertex is 0, the vertex indeed lies on the x-axis.

5. **Interpret the Meaning for the Quadratic Equation**:
   The question asks about the solutions to the equation:
   \[
   -x^2 + 4x - 4 = 0
   \]
   Since the vertex lies on the x-axis, this means the parabola touches the x-axis at exactly one point (the vertex), implying that this quadratic equation has only **one solution**.

### Answer:
The correct choice is **one** solution.

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### Related Questions:
1. How do you find the vertex of a parabola given its equation in standard form?
2. What does it mean geometrically if a quadratic equation has only one solution?
3. How can you determine the number of solutions for a quadratic equation without graphing it?
4. What is the significance of the discriminant in determining the number of solutions?
5. How does the sign of the leading coefficient affect the orientation of a parabola?

### Tip:
For a quadratic equation \( ax^2 + bx + c = 0 \), if the discriminant \( b^2 - 4ac \) is zero, the equation has exactly one solution (the parabola touches the x-axis at one point).

If the vertex of y = -x^2 + 4x - 4 lies on the x-axis, then the quadratic equation -x^2 + 4x - 4 = 0 has ____ solution(s).

Learn how to determine the number of solutions for a quadratic equation when the vertex lies on the x-axis. This solution uses algebraic concepts, including the vertex formula and the discriminant, to find that the equation has one solution. Suitable for high school students in Grades 9-11.

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