The value of \(\sqrt{12+\sqrt{12+\sqrt{12+\dots}}}\) is

Solution: Let x = √(12 + √(12 + √(12 + ...))).

• Assume the nested radical converges to a number x, so x = √(12 + x).

• Square both sides to eliminate the square root: x2 = 12 + x, which gives the quadratic equation x2 − x − 12 = 0.

• Solve the quadratic: x = [1 ± √(1 + 48)]/2 = [1 ± 7]/2, so the roots are x = 4 and x = −3.

• The nested radical is a square root and therefore nonnegative, so discard −3 and accept x = 4.

Answer: 4