Find the value of
`sqrt(72+sqrt(72+sqrt(72+.....oo`

Options

A. 6
B. 8
C. 9
D. 12

Answer

To find the value of the expression \( \sqrt{72 + \sqrt{72 + \sqrt{72 + \ldots}}} \), we can follow these steps:

### Step 1: Set up the equation
Let \( x \) be the value of the infinite nested radical. Therefore, we can write:
\[ x = \sqrt{72 + \sqrt{72 + \sqrt{72 + \ldots}}} \]
This implies:
\[ x = \sqrt{72 + x} \]

### Step 2: Square both sides
To eliminate the square root, we square both sides of the equation:
\[ x^2 = 72 + x \]

### Step 3: Rearrange the equation
Rearranging the equation gives us:
\[ x^2 - x - 72 = 0 \]

### Step 4: Solve the quadratic equation
Now we can solve the quadratic equation using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -1 \), and \( c = -72 \). Plugging in these values, we get:
\[ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-72)}}{2 \cdot 1} \]
\[ x = \frac{1 \pm \sqrt{1 + 288}}{2} \]
\[ x = \frac{1 \pm \sqrt{289}}{2} \]
\[ x = \frac{1 \pm 17}{2} \]

### Step 5: Calculate the possible values
Calculating the two possible values:
1. \( x = \frac{18}{2} = 9 \)
2. \( x = \frac{-16}{2} = -8 \)

Since \( x \) represents a length (as it is derived from a square root), we discard the negative value.

### Final Answer
Thus, the value of \( \sqrt{72 + \sqrt{72 + \sqrt{72 + \ldots}}} \) is:
\[ \boxed{9} \]