How to solve $x+\sqrt{1+x^2}=3\sqrt{3}$ for $x$?

Question

How do I solve the following equation for x:

$x+\sqrt{1+x^2}=3\sqrt{3}$

I'm failing miserably in isolating the $x$

Answer 1

Writing your equation in the form

$$\sqrt{1+x^2}=3\sqrt{3}-x$$ and by squaring we get $$1+x^2=27+x^2-6\sqrt{3}x$$ so we get $$6\sqrt{3}x=26$$ Can you finish?

Answer 2

Hint $$(\sqrt{1+x^2}+x)(\sqrt{1+x^2}-x)=1$$

$$\implies\sqrt{1+x^2}-x=\dfrac1{3\sqrt3}$$

Can you take it from here?