`sqrt(12+sqrt(12+sqrt(12+….)))` is equal to

Options

A. 3
B. 4
C. 6
D. 2

Answer

To solve the expression \( \sqrt{12 + \sqrt{12 + \sqrt{12 + \ldots}}} \), we can set it equal to a variable, say \( x \). This gives us the equation:

\[ x = \sqrt{12 + x} \]

### Step 1: Square both sides
To eliminate the square root, we square both sides of the equation:

\[ x^2 = 12 + x \]

### Step 2: Rearrange the equation
Next, we rearrange the equation to bring all terms to one side:

\[ x^2 - x - 12 = 0 \]

### Step 3: Factor the quadratic equation
Now, we need to factor the quadratic equation. We look for two numbers that multiply to \(-12\) and add up to \(-1\). The numbers \(-4\) and \(3\) fit this requirement:

\[ (x - 4)(x + 3) = 0 \]

### Step 4: Solve for \( x \)
Setting each factor equal to zero gives us the potential solutions:

\[ x - 4 = 0 \quad \Rightarrow \quad x = 4 \]
\[ x + 3 = 0 \quad \Rightarrow \quad x = -3 \]

### Step 5: Determine the valid solution
Since \( x \) represents a square root, it must be non-negative. Therefore, we discard \( x = -3 \) and keep:

\[ x = 4 \]

Thus, the value of \( \sqrt{12 + \sqrt{12 + \sqrt{12 + \ldots}}} \) is:

\[ \boxed{4} \]