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 $x$

Welcome to Math Stack Exchange. Isolate $\sqrt{1+x^2}$ on one side of the equation and then square both sides — J. W. Tanner, Jun 21 '19 at 13:29
So simple when you see it... I've tried everything except the right one! — thatOldITGuy, Jun 21 '19 at 13:35
Note that if you are squaring equations to solve them you need to take care. $x=a$ has one solution, but $x^2=a^2$ has two. So you do need here to show in some way that the value you obtain solves the original equation. It isn't difficult here, but in more complex questions, don't let it trip you up. (here the second solution disappears - informally, we might say to $x=-\infty$) — Mark Bennet, Jun 21 '19 at 13:56

score 2 · Accepted Answer · answered Jun 21 '19 at 13:30

2

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?

answered Jun 21 '19 at 13:30

I surely can. So simple after seeing it that is embarassing. – thatOldITGuy Jun 21 '19 at 13:36

score 1 · Answer 2 · answered Jun 21 '19 at 14:16

1

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?

answered Jun 21 '19 at 14:16

lab bhattacharjee

2 Answers2