Nested Radicals (1)

Let $x$ be the given number. Then we have:

$\begin{aligned} x & = \sqrt{12+\sqrt{12+ \sqrt{12+\cdots}}} \\ & = \sqrt{12 + x} & \small \color{#3D99F6} \text{squaring both sides} \\ x^2 & = 12 +x \\ x^2 - x -12 & = 0 \\ (x-4)(x+3) & = 0 \\ \implies x & = \boxed{4} & \small \color{#3D99F6} \text{Since }x > 0 \end{aligned}$

Let $x=\sqrt{12+\sqrt{12+\sqrt{12+\dots}}}$

Then $x^2-12=\sqrt{12+\sqrt{12+\sqrt{12+\dots}}}=x;x=4,-3$ , but $-3$ doesn't fit because it can't be a negative number.