Find the value of `sqrt(6+sqrt(6+sqrt(6+...oo)))`

Answer

To find the value of \( \sqrt{6 + \sqrt{6 + \sqrt{6 + \ldots}}} \), we can follow these steps:

### Step 1: Set up the equation
Let \( y = \sqrt{6 + \sqrt{6 + \sqrt{6 + \ldots}}} \). This means that \( y \) is equal to the entire expression.

### Step 2: Square both sides
To eliminate the square root, we square both sides of the equation:
\[ y^2 = 6 + \sqrt{6 + \sqrt{6 + \ldots}} \]
Since the right side is the same as \( y \), we can rewrite the equation as:
\[ y^2 = 6 + y \]

### Step 3: Rearrange the equation
Now, rearranging the equation gives us a standard quadratic form:
\[ y^2 - y - 6 = 0 \]

### Step 4: Apply the quadratic formula
To solve for \( y \), we can use the quadratic formula:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our equation, \( a = 1 \), \( b = -1 \), and \( c = -6 \). Plugging these values into the formula gives:
\[ y = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-6)}}{2 \cdot 1} \]
This simplifies to:
\[ y = \frac{1 \pm \sqrt{1 + 24}}{2} \]
\[ y = \frac{1 \pm \sqrt{25}}{2} \]

### Step 5: Simplify the square root
Since \( \sqrt{25} = 5 \), we have:
\[ y = \frac{1 \pm 5}{2} \]

### Step 6: Calculate the possible values of \( y \)
This gives us two possible solutions:
1. \( y = \frac{1 + 5}{2} = \frac{6}{2} = 3 \)
2. \( y = \frac{1 - 5}{2} = \frac{-4}{2} = -2 \)

### Step 7: Determine the valid solution
Since \( y \) represents a length (the value of the square root), it must be non-negative. Therefore, we discard \( y = -2 \).

Thus, the value of \( \sqrt{6 + \sqrt{6 + \sqrt{6 + \ldots}}} \) is:
\[ \boxed{3} \]