Evaluate the infinite nested radical x = √(12 + √(12 + √(12 + …))). Find the exact value of x.

Introduction / Context: Infinite nested radicals can often be evaluated by setting the entire expression equal to a variable and exploiting self-similarity. This tests algebraic reasoning and solving a quadratic equation arising from the structure.

Given Data / Assumptions:

• x = √(12 + √(12 + √(12 + …))).
• Assume convergence to a finite positive x (standard for such forms with positive constants).

Concept / Approach: Because the structure repeats, set x = √(12 + x). Then square both sides and rearrange to a quadratic in x. Solve for positive x (as square roots yield nonnegative values).

Step-by-Step Solution: Let x = √(12 + x). Square: x^2 = 12 + x. Rearrange: x^2 − x − 12 = 0. Solve quadratic: x = [1 ± √(1 + 48)] / 2 = [1 ± 7] / 2. Possible roots: x = 4 or x = −3. Since x is a value of a square root expression, x ≥ 0 ⇒ x = 4.

Verification / Alternative check: Substitute x = 4 back: √(12 + 4) = √16 = 4, which satisfies the self-referential equation.

Why Other Options Are Wrong: 3 and 6 do not satisfy x = √(12 + x). “Greater than 6” contradicts the exact solution x = 4.

Common Pitfalls: Including the negative root from the quadratic or attempting to expand the radical indefinitely rather than using self-similarity.

Final Answer: 4