Equation of axis or parabola `y^(2)+3y+2x+5=0` is

Options

A. x + 5 = 0
B. y+ 5 = 0
C. 2y + 3 = 0
D. y + 3 = 0

Answer

To find the equation of the axis of the parabola given by the equation \( y^2 + 3y + 2x + 5 = 0 \), we can follow these steps:

### Step 1: Rearrange the equation
Start by rearranging the given equation to isolate the terms involving \(x\):
\[ y^2 + 3y + 5 = -2x \]

### Step 2: Complete the square for the \(y\) terms
To complete the square for the \(y\) terms, we take the coefficient of \(y\), which is 3, divide it by 2 to get \( \frac{3}{2} \), and then square it to get \( \left(\frac{3}{2}\right)^2 = \frac{9}{4} \). We add and subtract this value:
\[ y^2 + 3y + \frac{9}{4} - \frac{9}{4} + 5 = -2x \]
This simplifies to:
\[ \left(y + \frac{3}{2}\right)^2 - \frac{9}{4} + 5 = -2x \]

### Step 3: Simplify the equation
Now simplify the constant terms:
\[ \left(y + \frac{3}{2}\right)^2 - \frac{9}{4} + \frac{20}{4} = -2x \]
This gives:
\[ \left(y + \frac{3}{2}\right)^2 + \frac{11}{4} = -2x \]

### Step 4: Rearranging to standard form
Rearranging gives:
\[ \left(y + \frac{3}{2}\right)^2 = -2x - \frac{11}{4} \]
This can be rewritten as:
\[ \left(y + \frac{3}{2}\right)^2 = -2\left(x + \frac{11}{8}\right) \]

### Step 5: Identify the vertex and axis
From the equation \(\left(y - k\right)^2 = 4p(x - h)\), we can identify that:
- The vertex is at \(\left(-\frac{11}{8}, -\frac{3}{2}\right)\)
- The axis of the parabola is given by the line \(y = k\), where \(k = -\frac{3}{2}\).

### Step 6: Write the equation of the axis
Thus, the equation of the axis is:
\[ y + \frac{3}{2} = 0 \quad \Rightarrow \quad 2y + 3 = 0 \]

### Final Answer
The equation of the axis of the parabola is:
\[ \boxed{2y + 3 = 0} \]