The value of `sqrt(72+sqrt(72+sqrt(72+)))` is

Options

A. 9
B. 8
C. 18
D. 12

Answer

To solve the problem \( \sqrt{72 + \sqrt{72 + \sqrt{72 + \ldots}}} \), we can follow these steps:

### Step 1: Set up the equation
Let \( x = \sqrt{72 + \sqrt{72 + \sqrt{72 + \ldots}}} \). This means we can express the equation as:
\[ x = \sqrt{72 + x} \]

**Hint:** Define the infinite nested radical as a variable to simplify the equation.

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

**Hint:** Squaring both sides helps to remove the square root, allowing us to work with a polynomial equation.

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

**Hint:** Move all terms to one side to form a standard quadratic equation.

### Step 4: Factor the quadratic equation
Next, we need to factor the quadratic equation. We look for two numbers that multiply to \(-72\) and add to \(-1\). The factors are:
\[ (x - 9)(x + 8) = 0 \]

**Hint:** Factoring the quadratic helps us find the possible values of \(x\).

### Step 5: Solve for \(x\)
Setting each factor equal to zero gives us:
1. \(x - 9 = 0 \Rightarrow x = 9\)
2. \(x + 8 = 0 \Rightarrow x = -8\)

Since \(x\) represents a square root, it must be non-negative. Therefore, we discard \(x = -8\).

**Hint:** Consider the context of the problem; negative values are not valid for square roots.

### Step 6: Conclusion
Thus, the value of \( \sqrt{72 + \sqrt{72 + \sqrt{72 + \ldots}}} \) is:
\[ \boxed{9} \]

**Hint:** Always check if the solution fits the original equation to ensure it is valid.