What proofs are there of this result (on square roots and continued fractions)?

Question

$ \sqrt{a^2+b} = a+\cfrac{b}{2a+\cfrac{b}{2a+\cfrac{b}{2a+\dots}}}$

For example $\sqrt{7} \approx 2.64575$ Here we have :

$\sqrt{2^2+3} = 2+\cfrac{3}{4+\cfrac{3}{4+\cfrac{3}{4+\dots}}}$

According to the book Continued Fractions By C. D. Olds Continued Fractions, this result would have been first discovered by Rafael Bombelli in his 1752 treaty on algebra.

Answer 1

Thanks to Raymond Manzoni's comment I understand now, for $x >0$

$x = a + \frac{b}{a+x} \iff x^2 = a^2 + b$ and so we have $x = \sqrt{a^2+b}$