Can You Eyeball This?

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

$\sqrt{x^2+1} = n-\sqrt{x+1}$

Squaring both sides,

$x^2+1= n^2+x+1-2n \sqrt{x+1}$

$\implies \sqrt{x+1} = \dfrac{ x^2-n^2-x}{-2n}$

Squaring both sides and putting $n=7\sqrt{2}$ ,

$\implies x+1=\dfrac{(x^2-98-x)^2}{392}$

This gives us

$x^4-2x^3-195x^2-196x+9212=0$

$\Rightarrow x= \boxed{\color{#E81990}{\boxed{\color{#3D99F6}{7}}}}$

[This is not yet a solution. If no one adds a non-trial and error solution in a week, I will flesh out the details.]

So, most people might eventually guess that $x = 7$ gives us $\sqrt{ x + 1 } + \sqrt{ x^2 + 1 } = \sqrt{ 8 } + \sqrt{ 50 } = 7 \sqrt{2 }$ .

However, this is somewhat haphazard, and doesn't allow us to solve the more general problem of $\sqrt{ x + 1 } + \sqrt{ x^2 + 1 } = n$ .

Do you see how we could form a polynomial equation to figure out the value of $n$ ?