- #1

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## 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.