Find the value of `sqrt(12-sqrt(12-sqrt(12-...)))`

Answer

To solve the expression \( \sqrt{12 - \sqrt{12 - \sqrt{12 - \ldots}}} \), we can follow these steps:

### Step 1: Set Up the Equation
Let \( x = \sqrt{12 - \sqrt{12 - \sqrt{12 - \ldots}}} \). This means we can rewrite the equation as:
\[ x = \sqrt{12 - x} \]

### Step 2: Square Both Sides
To eliminate the square root, we square both sides of the equation:
\[ x^2 = 12 - x \]

### Step 3: Rearrange the Equation
Rearranging the equation gives us:
\[ x^2 + x - 12 = 0 \]

### Step 4: Factor the Quadratic Equation
Now we need to factor the quadratic equation. We look for two numbers that multiply to \(-12\) and add to \(1\). The numbers \(4\) and \(-3\) work:
\[ (x + 4)(x - 3) = 0 \]

### Step 5: Solve for \( x \)
Setting each factor to zero gives us the possible solutions:
1. \( x + 4 = 0 \) → \( x = -4 \)
2. \( x - 3 = 0 \) → \( x = 3 \)

### Step 6: Determine the Valid Solution
Since \( x \) represents a square root, it must be non-negative. Therefore, we discard \( x = -4 \) and keep:
\[ x = 3 \]

### Conclusion
Thus, the value of \( \sqrt{12 - \sqrt{12 - \sqrt{12 - \ldots}}} \) is:
\[ \boxed{3} \]