Infinite nested radical value: Evaluate √(30 + √(30 + √(30 + …))) to its exact finite value.

Introduction / Context:Infinite nested radicals often converge to a finite value x that satisfies a self-referential equation: x = √(constant + x). Solving the resulting quadratic yields the limit value. Only the positive root is meaningful because the radical is nonnegative.Given Data / Assumptions:

• Expression: x = √(30 + x).
• x ≥ 0 due to square roots.

Concept / Approach:Square both sides to eliminate the outer radical and form a quadratic in x. Solve and select the nonnegative solution consistent with the original expression.

Step-by-Step Solution:

Verification / Alternative check:Plug back: √(30 + 6) = √36 = 6, confirming consistency.

Why Other Options Are Wrong:

• 5, 7: Do not satisfy x = √(30 + x).
• 3√10 ≈ 9.486..., far from the correct fixed point.

Common Pitfalls:Keeping both quadratic roots without checking the radical’s nonnegativity constraint, or mishandling the square step.

Final Answer: