If `x=sqrt(6+sqrt(6+sqrt(6......)))`, then

Options

A. 3
B. 2
C. 15
D. `x-y+1=0`

Answer

To solve the equation \( x = \sqrt{6 + \sqrt{6 + \sqrt{6 + \ldots}}} \), we can follow these steps:

### Step 1: Set up the equation
We start with the equation given in the problem:
\[ x = \sqrt{6 + x} \]

### Step 2: Square both sides
To eliminate the square root, we square both sides of the equation:
\[ x^2 = 6 + x \]

### Step 3: Rearrange the equation
Next, we rearrange the equation to set it to zero:
\[ x^2 - x - 6 = 0 \]

### Step 4: Factor the quadratic equation
Now, we need to factor the quadratic equation. We look for two numbers that multiply to \(-6\) and add to \(-1\). The numbers \(-3\) and \(2\) work:
\[ (x - 3)(x + 2) = 0 \]

### Step 5: Solve for \(x\)
Setting each factor to zero gives us the potential solutions:
1. \(x - 3 = 0 \Rightarrow x = 3\)
2. \(x + 2 = 0 \Rightarrow x = -2\)

### Step 6: Determine the valid solution
Since \(x\) represents a square root, it must be non-negative. Therefore, we discard \(x = -2\) and accept:
\[ x = 3 \]

### Final Answer
Thus, the value of \(x\) is:
\[ \boxed{3} \]