Question

Express as a continued fraction the positive root of the following equation $x^2-4x-3=0$.

Answer 1

$x^2-4x-3=0$

$\Rightarrow x=\frac{4\pm \sqrt{28}}{2}$

$2\pm \sqrt{7}$

So, the positive root of the equation is $(2+\sqrt{7})$

Now, $2+\sqrt{7}=3+(\sqrt{7}-1)$

$=3+\frac{6}{1+\sqrt{7}}$

$=3+\frac{6}{1+1+\frac{6}{1+\sqrt{7}}}$

$=3+\frac{6}{2+\frac{6}{1+\frac{1}{1+\sqrt{7}}}}$

$=1+\frac{6}{2+\frac{6}{2+\frac{6}{2+...}}}$