`(sqrt(56+sqrt(56+sqrt(56+...))))/2^(2)=?`

Options

A. 0
B. 1
C. 2
D. 8

Answer

To solve the expression \(\frac{\sqrt{56 + \sqrt{56 + \sqrt{56 + \ldots}}}}{2^2}\), we can follow these steps:

### Step 1: Define the Infinite Nested Radical
Let \( p = \sqrt{56 + \sqrt{56 + \sqrt{56 + \ldots}}} \). This means that \( p \) can be expressed as:
\[ p = \sqrt{56 + p} \]

### Step 2: Square Both Sides
To eliminate the square root, we square both sides:
\[ p^2 = 56 + p \]

### Step 3: Rearrange the Equation
Rearranging this equation gives us:
\[ p^2 - p - 56 = 0 \]

### Step 4: Solve the Quadratic Equation
Now we can solve this quadratic equation using the quadratic formula \( p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = -1, c = -56 \):
\[ p = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-56)}}{2 \cdot 1} \]
\[ p = \frac{1 \pm \sqrt{1 + 224}}{2} \]
\[ p = \frac{1 \pm \sqrt{225}}{2} \]
\[ p = \frac{1 \pm 15}{2} \]

### Step 5: Calculate the Possible Values of \( p \)
Calculating the two possible values:
1. \( p = \frac{16}{2} = 8 \)
2. \( p = \frac{-14}{2} = -7 \) (we discard this since \( p \) must be positive)

Thus, we have:
\[ p = 8 \]

### Step 6: Substitute Back into the Original Expression
Now we substitute \( p \) back into the original expression:
\[ \frac{p}{2^2} = \frac{8}{4} = 2 \]

### Final Answer
The final answer is:
\[ \boxed{2} \]