Find the value of `sqrt(30-sqrt(30-sqrt(30...)))`

Answer

To solve the expression \( \sqrt{30 - \sqrt{30 - \sqrt{30 - \ldots}}} \), we can follow these steps:

### Step 1: Set up the equation
Let \( x = \sqrt{30 - \sqrt{30 - \sqrt{30 - \ldots}}} \). This means we can rewrite the expression as:
\[ x = \sqrt{30 - x} \]

### Step 2: Square both sides
To eliminate the square root, we square both sides of the equation:
\[ x^2 = 30 - x \]

### Step 3: Rearrange the equation
Now, we rearrange the equation to bring all terms to one side:
\[ x^2 + x - 30 = 0 \]

### Step 4: Factor the quadratic equation
Next, we need to factor the quadratic equation. We look for two numbers that multiply to \(-30\) and add to \(1\). The numbers \(6\) and \(-5\) work:
\[ (x + 6)(x - 5) = 0 \]

### Step 5: Solve for \(x\)
Setting each factor equal to zero gives us the possible solutions:
\[ x + 6 = 0 \quad \Rightarrow \quad x = -6 \]
\[ x - 5 = 0 \quad \Rightarrow \quad x = 5 \]

### Step 6: Determine the valid solution
Since \(x\) represents a square root, it must be non-negative. Therefore, we discard \(x = -6\) and accept:
\[ x = 5 \]

### Conclusion
The value of \( \sqrt{30 - \sqrt{30 - \sqrt{30 - \ldots}}} \) is \( \boxed{5} \).

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