If `x = sqrt(6 + sqrt(6 + sqrt(6 + sqrt(6 + …oo))))`, then what is one of the values of x equal to?

Options

A. 6
B. 5
C. 4
D. 3

Answer

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

### Step 1: Set up the equation
We start by recognizing that the expression inside the square root continues indefinitely. Therefore, we can set:
\[ x = \sqrt{6 + x} \]

### 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 bring all terms to one side:
\[ x^2 - x - 6 = 0 \]

### Step 4: Factor the quadratic equation
Now, we will factor the quadratic equation. We need 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 equal to zero gives us the possible solutions:
\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \]
\[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \]

### Step 6: Determine the valid solution
Since \(x\) represents a length (as it is derived from a square root), we discard the negative solution. Thus, we have:
\[ x = 3 \]

### Conclusion
One of the values of \(x\) is equal to \(3\).

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