For real solution of equation `3sqrt(x+3p+1)-3sqrt(x)=1`, we have

Options

A. `p ge 1//4`
B. `p ge -1//4`
C. `p ge 1//3`
D. `p ge -1//3`

Explanation

To solve the equation \( 3\sqrt{x + 3p + 1} - 3\sqrt{x} = 1 \) for real solutions, we will follow these steps:

### Step 1: Isolate the square root
We start by isolating one of the square root terms:

\[ 3\sqrt{x + 3p + 1} = 3\sqrt{x} + 1 \]

### Step 2: Divide by 3
Next, we divide both sides of the equation by 3:

\[ \sqrt{x + 3p + 1} = \sqrt{x} + \frac{1}{3} \]

### Step 3: Square both sides
Now we will square both sides to eliminate the square roots:

\[ x + 3p + 1 = \left(\sqrt{x} + \frac{1}{3}\right)^2 \]

### Step 4: Expand the right side
Expanding the right side gives:

\[ x + 3p + 1 = x + \frac{2}{3}\sqrt{x} + \frac{1}{9} \]

### Step 5: Rearranging the equation
Now, we rearrange the equation to isolate the square root:

\[ 3p + 1 - \frac{1}{9} = \frac{2}{3}\sqrt{x} \]

### Step 6: Simplify the left side
We simplify the left side:

\[ 3p + \frac{8}{9} = \frac{2}{3}\sqrt{x} \]

### Step 7: Isolate \(\sqrt{x}\)
Now, we isolate \(\sqrt{x}\):

\[ \sqrt{x} = \frac{3}{2}(3p + \frac{8}{9}) \]

### Step 8: Square both sides again
Square both sides again to eliminate the square root:

\[ x = \left(\frac{3}{2}(3p + \frac{8}{9})\right)^2 \]

### Step 9: Set up the quadratic equation
We can express this in the form of a quadratic equation. Let \( h = \sqrt{x} \):

\[ h^2 + h - p = 0 \]

### Step 10: Determine the discriminant
For the quadratic equation \( h^2 + h - p = 0 \) to have real solutions, the discriminant must be non-negative:

\[ D = b^2 - 4ac = 1^2 - 4(1)(-p) = 1 + 4p \geq 0 \]

### Step 11: Solve for \( p \)
This leads to the condition:

\[ 1 + 4p \geq 0 \implies p \geq -\frac{1}{4} \]

### Conclusion
Thus, for the real solution of the equation \( 3\sqrt{x + 3p + 1} - 3\sqrt{x} = 1 \), we have:

\[ p \geq -\frac{1}{4} \]

This corresponds to option 2.

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