How can continued fractions be used to prove a deep result in mathematics?

[DeadOriginal]

Homework Statement

Let x be any positive real number and suppose that $x^2-ax-b=0$ where $a,b$ are positive. I would like to use the equation that I provided in relevant equations which I proved to prove that
$$
\sqrt{\alpha^{2}+\beta}=\alpha+\cfrac{\beta}{2 \alpha+\cfrac{\beta}{2 \alpha+\cfrac{\beta}{2\alpha+\ddots}}}
$$
where $\alpha,\beta>0$.

Homework Equations

I proved that
$$
x=a+\cfrac{b}{a+\cfrac{b}{a+\cfrac{b}{a+\ddots}}}.
$$

The Attempt at a Solution

I tried to do things like find values of $a,b$ so that when I transformed the equation $x^{2}-ax-b=0$ into a continued fraction that I would get the desired continued fraction with $x=\sqrt{\alpha^{2}+\beta}$ but that didn't work out.

I also tried changing $\sqrt{\alpha^{2}+\beta}$ directly into a continued fraction using the canonical continued fraction algorithm but I then had to consider different values of $\beta$ which would give me different continued fractions that I didn't really know how to combine to create the desired continued fraction.

I tried to plug $x=\sqrt{\alpha^{2}+\beta}$ into $x^{2}-ax-b=0$ and then solve for $a,b$ but that didn't get too far with two variables and one equation.

 

[haruspex]

If you set a = 2α and b = β, what are the roots of x²-ax-b?

 

[DeadOriginal]

If you set a = 2α and b = β, what are the roots of x²-ax-b?

I don't know why but I remember trying that and it didn't work but now that I try it again after you mention it, it works. Thanks!

 

[arildno]

I don't know why but I remember trying that and it didn't work but now that I try it again after you mention it, it works. Thanks!

A deep result in maths is that arithmetic and algebraic results are NOT static over time, but depend on what help you get!

 

[DeadOriginal]

A deep result in maths is that arithmetic and algebraic results are NOT static over time, but depend on what help you get!

Amen! Haha.