What is the value of `y=sqrt(8+2sqrt(8+2sqrt(8+2sqrt(8+oo))))`
(a)10
(b)8
(c)6
(d)4

Options

A. 10
B. 8
C. 6
D. 4

Answer

To solve the equation \( y = \sqrt{8 + 2\sqrt{8 + 2\sqrt{8 + 2\sqrt{8 + \infty}}}} \), we can simplify it step by step.

### Step 1: Set up the equation
We start by recognizing that the expression inside the square root is recursive. We can set:
\[ y = \sqrt{8 + 2y} \]

### Step 2: Square both sides
To eliminate the square root, we square both sides of the equation:
\[ y^2 = 8 + 2y \]

### Step 3: Rearrange the equation
Next, we rearrange the equation to bring all terms to one side:
\[ y^2 - 2y - 8 = 0 \]

### Step 4: Factor the quadratic equation
Now, we need to factor the quadratic equation. We look for two numbers that multiply to \(-8\) and add to \(-2\). The numbers \(-4\) and \(2\) work:
\[ (y - 4)(y + 2) = 0 \]

### Step 5: Solve for \(y\)
Setting each factor equal to zero gives us:
1. \( y - 4 = 0 \) → \( y = 4 \)
2. \( y + 2 = 0 \) → \( y = -2 \)

Since \(y\) represents a square root, we discard the negative solution:
\[ y = 4 \]

### Conclusion
Thus, the value of \( y \) is \( 4 \), which corresponds to option (d).