Calculate a quadratic irrational from its periodic continued fraction

Question

I have a periodic continued fraction [2; 1, 3] and I want to convert it into a quadratic irrational.

Any helps?

Answer 1

Quick answer: From link, you can use the formula $$[a,b] = \frac{-ab+\sqrt{ab(ab+4)}}{2a}.$$

Then, you can check the result here with $\frac{1+\sqrt{21}}{2}$

Answer 2

$x=(1;\overline{3,1})\implies x=1+\cfrac1{3+\cfrac1x}\implies3x^2-3x-1=0$. Thus, $$ (1;\overline{3,1})=\frac{3+\sqrt{21}}6 $$ Therefore, $$ \begin{align} (2;\overline{1,3}) &=2+\frac6{3+\sqrt{21}}\\[3pt] &=\frac{1+\sqrt{21}}2 \end{align} $$

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$x=(3;\overline{3})\implies x=3+\cfrac1x\implies x^2-3x-1=0$. Thus, $$ (3;\overline{3})=\frac{3+\sqrt{13}}2 $$ Therefore, $$ (2;1,\overline{3})=\frac{13+\sqrt{13}}6 $$