$$\left. \begin{array} { l } { a = 1 - \sqrt { 5 } } \\ { a ^ { 2 } - \frac { 1 } { a ^ { 2 } } } \end{array} \right.$$
$b=\frac{45-17\sqrt{5}}{8}\approx 0.873355548$
$$b=\left(1-\sqrt{5}\right)^{2}-\frac{1}{\left(1-\sqrt{5}\right)^{2}}$$
$$b=1-2\sqrt{5}+\left(\sqrt{5}\right)^{2}-\frac{1}{\left(1-\sqrt{5}\right)^{2}}$$
$$b=1-2\sqrt{5}+5-\frac{1}{\left(1-\sqrt{5}\right)^{2}}$$
$$b=6-2\sqrt{5}-\frac{1}{\left(1-\sqrt{5}\right)^{2}}$$
$$b=6-2\sqrt{5}-\frac{1}{1-2\sqrt{5}+\left(\sqrt{5}\right)^{2}}$$
$$b=6-2\sqrt{5}-\frac{1}{1-2\sqrt{5}+5}$$
$$b=6-2\sqrt{5}-\frac{1}{6-2\sqrt{5}}$$
$$b=6-2\sqrt{5}-\frac{6+2\sqrt{5}}{\left(6-2\sqrt{5}\right)\left(6+2\sqrt{5}\right)}$$
$$b=6-2\sqrt{5}-\frac{6+2\sqrt{5}}{6^{2}-\left(-2\sqrt{5}\right)^{2}}$$
$$b=6-2\sqrt{5}-\frac{6+2\sqrt{5}}{36-\left(-2\sqrt{5}\right)^{2}}$$
$$b=6-2\sqrt{5}-\frac{6+2\sqrt{5}}{36-\left(-2\right)^{2}\left(\sqrt{5}\right)^{2}}$$
$$b=6-2\sqrt{5}-\frac{6+2\sqrt{5}}{36-4\left(\sqrt{5}\right)^{2}}$$
$$b=6-2\sqrt{5}-\frac{6+2\sqrt{5}}{36-4\times 5}$$
$$b=6-2\sqrt{5}-\frac{6+2\sqrt{5}}{36-20}$$
$$b=6-2\sqrt{5}-\frac{6+2\sqrt{5}}{16}$$
$$b=6-2\sqrt{5}-\left(\frac{3}{8}+\frac{1}{8}\sqrt{5}\right)$$
$$b=6-2\sqrt{5}-\frac{3}{8}-\frac{1}{8}\sqrt{5}$$
$$b=\frac{45}{8}-2\sqrt{5}-\frac{1}{8}\sqrt{5}$$
$$b=\frac{45}{8}-\frac{17}{8}\sqrt{5}$$
$$a=1-\sqrt{5}$$ $$b=\frac{45}{8}-\frac{17}{8}\sqrt{5}$$
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