Find the vertex, focus and directrix of the parabola `x^(2)=2(2x+y)`.

Explanation

To find the vertex, focus, and directrix of the parabola given by the equation \( x^2 = 2(2x + y) \), we will follow these steps:

### Step 1: Rewrite the equation
Start with the given equation:
\[ x^2 = 2(2x + y) \]
Distributing the right side:
\[ x^2 = 4x + 2y \]

### Step 2: Rearrange the equation
Rearranging the equation to isolate \( y \):
\[ x^2 - 4x = 2y \]
Now divide both sides by 2:
\[ y = \frac{1}{2}(x^2 - 4x) \]
This can be rewritten as:
\[ y = \frac{1}{2}(x^2 - 4x) = \frac{1}{2}(x^2 - 4x + 4 - 4) = \frac{1}{2}((x - 2)^2 - 4) \]
Thus, we have:
\[ y = \frac{1}{2}(x - 2)^2 - 2 \]

### Step 3: Identify the vertex
From the standard form of the parabola \( y = a(x - h)^2 + k \), we can identify the vertex \((h, k)\):
- Here, \( h = 2 \) and \( k = -2 \).
- Therefore, the vertex is:
\[ (2, -2) \]

### Step 4: Determine the value of \( a \)
From the equation, we see that \( a = \frac{1}{2} \).

### Step 5: Find the focus
The focus of a parabola is located at a distance \( a \) above the vertex along the axis of symmetry. Since the vertex is at \( (2, -2) \) and \( a = \frac{1}{2} \):
- The focus is at:
\[ (2, -2 + \frac{1}{2}) = (2, -\frac{3}{2}) \]

### Step 6: Find the directrix
The directrix is located at a distance \( a \) below the vertex. Thus, the directrix is:
\[ y = -2 - \frac{1}{2} = -\frac{5}{2} \]

### Summary of results
- **Vertex**: \( (2, -2) \)
- **Focus**: \( (2, -\frac{3}{2}) \)
- **Directrix**: \( y = -\frac{5}{2} \)