Root question help needed

Question

$$\sqrt{3+\sqrt{3+\sqrt{3+x}}}=x$$

Question is: How to find x?

Could you help me? Thanks in advance

Answer 1

Check this sly trick out. Just keep plugging in for $x$.

$$\begin{align*} x&=\sqrt{3+x}\\ &=\sqrt{3+\sqrt{3+x}}\\ &=\sqrt{3+\sqrt{3+\sqrt{3+x}}}\\ &=\sqrt{3+\sqrt{3+\sqrt{3+\cdots}}} \end{align*}$$

This recursive behavior implies that

$$\sqrt{3+\sqrt{3+\sqrt{3+x}}}=\sqrt{3+x}$$

So just cut yourself some slack and solve the equation

$$x=\sqrt{3+x}$$