If `y=sqrt(42-sqrt(42-sqrt(42.....oo)))` then y=?

Options

A. 7
B. 6
C. 5
D. 8

Answer

To solve the equation \( y = \sqrt{42 - \sqrt{42 - \sqrt{42 - \ldots}}} \), we can follow these steps:

### Step 1: Set up the equation
We start by recognizing that the expression inside the square root is the same as \( y \). Therefore, we can rewrite the equation as:
\[ y = \sqrt{42 - y} \]

### Step 2: Square both sides
To eliminate the square root, we square both sides of the equation:
\[ y^2 = 42 - y \]

### Step 3: Rearrange the equation
Next, we rearrange the equation to bring all terms to one side:
\[ y^2 + y - 42 = 0 \]

### Step 4: Factor the quadratic equation
Now, we need to factor the quadratic equation. We look for two numbers that multiply to \(-42\) and add to \(1\). The numbers \(7\) and \(-6\) work:
\[ (y + 7)(y - 6) = 0 \]

### Step 5: Solve for \(y\)
Setting each factor equal to zero gives us:
\[ y + 7 = 0 \quad \text{or} \quad y - 6 = 0 \]
This results in:
\[ y = -7 \quad \text{or} \quad y = 6 \]

### Step 6: Determine the valid solution
Since \(y\) represents a square root, it must be non-negative. Therefore, we discard \(y = -7\) and keep:
\[ y = 6 \]

### Final Answer
Thus, the value of \(y\) is:
\[ \boxed{6} \]